System and method for combining mimo and mode-division multiplexing

ABSTRACT

A communications system comprises first signal processing circuitry for receiving a plurality of input data streams and applying a different orthogonal function to each of the plurality of input data streams. Second signal processing circuitry processes each of the plurality of input data streams having the different orthogonal function applied thereto to multiplex a first group of the plurality of input data streams having a first group of orthogonal functions applied thereto onto a carrier signal and to multiplex a second group of the plurality of input data streams having a second group of orthogonal functions applied thereto onto the carrier signal. A MIMO transmitter transmits the carrier signal including the first group of the plurality of input data streams having the first group of orthogonal functions applied thereto and the second group of the plurality of input data streams having the second group of orthogonal functions applied thereto over a plurality of separate communications links. Each of the plurality of separate communications links go from one transmitting antenna of a plurality of transmitting antennas to each of a plurality of receiving antennas at a MIMO receiver.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No.15/216,474, filed on Jul. 21, 2016, entitled SYSTEM AND METHOD FORCOMBINING MIMO AND MODE-DIVISION MULTIPLEXING (Atty. Dkt. No.NXGN-33163), which U.S. application Ser. No. 15/216,474 claims benefitof U.S. Provisional Application No. 62/196,075, filed on Jul. 23, 2015,entitled SYSTEM AND METHOD FOR COMBINING MIMO AND MODE-DIVISIONMULTIPLEXING (Atty. Dkt. No. NXGN-32743). U.S. application Ser. No.15/216,474 is also a Continuation-In-Part of U.S. application Ser. No.14/882,085, filed on Oct. 13, 2015, entitled APPLICATION OF ORBITALANGULAR MOMENTUM TO FIBER, FSO AND RF (Atty. Dkt. No. NXGN-32777), whichU.S. application Ser. No. 14/882,085 claims benefit of U.S. ProvisionalApplication No. 62/063,028, filed Oct. 13, 2014, entitled APPLICATION OFORBITAL ANGULAR MOMENTUM TO FIBER, FSO AND RF (Atty. Dkt. No.NXGN-32392). U.S. patent application Ser. Nos. 15/216,474, 62/196,075,14/882,085 and 62/063,028 are incorporated by reference herein in theirentireties.

TECHNICAL FIELD

The following disclosure relates to systems and methods for increasingcommunication bandwidth, and more particularly to increasingcommunications bandwidth using a combination of mode divisionmultiplexing (MDM) and multiple input, multiple output transmissionsystems.

BACKGROUND

The use of voice and data networks has greatly increased as the numberof personal computing and communication devices, such as laptopcomputers, mobile telephones, Smartphones, tablets, et cetera, hasgrown. The astronomically increasing number of personal mobilecommunication devices has concurrently increased the amount of databeing transmitted over the networks providing infrastructure for thesemobile communication devices. As these mobile communication devicesbecome more ubiquitous in business and personal lifestyles, theabilities of these networks to support all of the new users and userdevices has been strained. Thus, a major concern of networkinfrastructure providers is the ability to increase their bandwidth inorder to support the greater load of voice and data communications andparticularly video that are occurring. Traditional manners forincreasing the bandwidth in such systems have involved increasing thenumber of channels so that a greater number of communications may betransmitted, or increasing the speed at which information is transmittedover existing channels in order to provide greater throughput levelsover the existing channel resources.

However, while each of these techniques have improved system bandwidths,existing technologies have taken the speed of communications to a levelsuch that drastic additional speed increases are not possible, eventhough bandwidth requirements due to increased usage are continuing togrow exponentially. Additionally, the number of channels assigned forvoice and data communications, while increasing somewhat, have notincreased to a level to completely support the increasing demands of avoice and data intensive use society. Thus, there is a great need forsome manner for increasing the bandwidth throughput within existingvoice and data communication that increases the bandwidth on existingvoice and data channels.

SUMMARY

The present invention, as disclosed and described herein, in at leastone aspect thereof, comprises a communications system having firstsignal processing circuitry for receiving a plurality of input datastreams and applying a different orthogonal function to each of theplurality of input data streams. Second signal processing circuitryprocesses each of the plurality of input data streams having thedifferent orthogonal function applied thereto to multiplex a first groupof the plurality of input data streams having a first group oforthogonal functions applied thereto onto a carrier signal and tomultiplex a second group of the plurality of input data streams having asecond group of orthogonal functions applied thereto onto the carriersignal. A MIMO transmitter transmits the carrier signal including thefirst group of the plurality of input data streams having the firstgroup of orthogonal functions applied thereto and the second group ofthe plurality of input data streams having the second group oforthogonal functions applied thereto over a plurality of separatecommunications links. Each of the plurality of separate communicationslinks go from one transmitting antenna of a plurality of transmittingantennas to each of a plurality of receiving antennas at a MIMOreceiver.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to thefollowing description taken in conjunction with the accompanyingDrawings in which:

FIG. 1 illustrates various techniques for increasing spectral efficiencywithin a transmitted signal;

FIG. 2 illustrates a particular technique for increasing spectralefficiency within a transmitted signal;

FIG. 3 illustrates a general overview of the manner for providingcommunication bandwidth between various communication protocolinterfaces;

FIG. 4 illustrates the manner for utilizing multiple level overlaymodulation with twisted pair/cable interfaces;

FIG. 5 illustrates a general block diagram for processing a plurality ofdata streams within an optical communication system;

FIG. 6 is a functional block diagram of a system for generating orbitalangular momentum within a communication system;

FIG. 7 is a functional block diagram of the orbital angular momentumsignal processing block of FIG. 6;

FIG. 8 is a functional block diagram illustrating the manner forremoving orbital angular momentum from a received signal including aplurality of data streams;

FIG. 9 illustrates a single wavelength having two quanti-spinpolarizations providing an infinite number of signals having variousorbital angular momentums associated therewith;

FIG. 10A illustrates an object with only a spin angular momentum;

FIG. 10B illustrates an object with an orbital angular momentum;

FIG. 10C illustrates a circularly polarized beam carrying spin angularmomentum;

FIG. 10D illustrates the phase structure of a light beam carrying anorbital angular momentum;

FIG. 11A illustrates a plane wave having only variations in the spinangular momentum;

FIG. 11B illustrates a signal having both spin and orbital angularmomentum applied thereto;

FIGS. 12A-12C illustrate various signals having different orbitalangular momentum applied thereto;

FIG. 12D illustrates a propagation of Poynting vectors for various Eigenmodes;

FIG. 12E illustrates a spiral phase plate;

FIG. 13 illustrates a system for using to the orthogonality of an HGmodal group for free space spatial multiplexing;

FIG. 14 illustrates a multiple level overlay modulation system;

FIG. 15 illustrates a multiple level overlay demodulator;

FIG. 16 illustrates a multiple level overlay transmitter system;

FIG. 17 illustrates a multiple level overlay receiver system;

FIGS. 18A-18K illustrate representative multiple level overlay signalsand their respective spectral power densities;

FIG. 19 illustrates comparisons of multiple level overlay signals withinthe time and frequency domain;

FIG. 20A illustrates a spectral alignment of multiple level overlaysignals for differing bandwidths of signals;

FIG. 20B-20C illustrate frequency domain envelopes located in separatelayers within a same physical bandwidth;

FIG. 21 illustrates an alternative spectral alignment of multiple leveloverlay signals;

FIG. 22 illustrates three different super QAM signals;

FIG. 23 illustrates the creation of inter-symbol interference inoverlapped multilayer signals;

FIG. 24 illustrates overlapped multilayer signals;

FIG. 25 illustrates a fixed channel matrix;

FIG. 26 illustrates truncated orthogonal functions;

FIG. 27 illustrates a typical OAM multiplexing scheme;

FIG. 28 illustrates various manners for converting a Gaussian beam intoan OAM beam;

FIG. 29A illustrates a fabricated metasurface phase plate;

FIG. 29B illustrates a magnified structure of the metasurface phaseplate;

FIG. 29C illustrates an OAM beam generated using the phase plate withl=+1;

FIG. 30 illustrates the manner in which a q-plate can convert a leftcircularly polarized beam into a right circular polarization orvice-versa;

FIG. 31 illustrates the use of a laser resonator cavity for producing anOAM beam;

FIG. 32 illustrates spatial multiplexing using cascaded beam splitters;

FIG. 33 illustrated de-multiplexing using cascaded beam splitters andconjugated spiral phase holograms;

FIG. 34 illustrates a log polar geometrical transformation based on OAMmultiplexing and de-multiplexing;

FIG. 35 illustrates an intensity profile of generated OAM beams andtheir multiplexing;

FIG. 36A illustrates the optical spectrum of each channel after eachmultiplexing for the OAM beams of FIG. 10A;

FIG. 36B illustrates the recovered constellations of 16-QAM signalscarried on each OAM beam;

FIG. 37A illustrates the steps to produce 24 multiplex OAM beams;

FIG. 37B illustrates the optical spectrum of a WDM signal carrier on anOAM beam;

FIG. 38A illustrates a turbulence emulator;

FIG. 38B illustrates the measured power distribution of an OAM beamafter passing through turbulence with a different strength;

FIG. 39A illustrates how turbulence effects mitigation using adaptiveoptics;

FIG. 39B illustrates experimental results of distortion mitigation usingadaptive optics;

FIG. 40 illustrates a free-space optical data link using OAM;

FIG. 41A illustrates simulated spot sized of different orders of OAMbeams as a function of transmission distance for a 3 cm transmittedbeam;

FIG. 41B illustrates simulated power loss as a function of aperturesize;

FIG. 42A illustrates a perfectly aligned system between a transmitterand receiver;

FIG. 42B illustrates a system with lateral displacement of alignmentbetween a transmitter and receiver;

FIG. 42C illustrates a system with receiver angular error for alignmentbetween a transmitter and receiver;

FIG. 43A illustrates simulated power distribution among different OAMmodes with a function of lateral displacement;

FIG. 43B illustrates simulated power distribution among different OAMmodes as a function of receiver angular error;

FIG. 44 illustrates a bandwidth efficiency comparison for square rootraised cosine versus multiple layer overlay for a symbol rate of ⅙;

FIG. 45 illustrates a bandwidth efficiency comparison between squareroot raised cosine and multiple layer overlay for a symbol rate of ¼;

FIG. 46 illustrates a performance comparison between square root raisedcosine and multiple level overlay using ACLR;

FIG. 47 illustrates a performance comparison between square root raisedcosine and multiple lever overlay using out of band power;

FIG. 48 illustrates a performance comparison between square root raisedcosine and multiple lever overlay using band edge PSD;

FIG. 49 is a block diagram of a transmitter subsystem for use withmultiple level overlay;

FIG. 50 is a block diagram of a receiver subsystem using multiple leveloverlay;

FIG. 51 illustrates an equivalent discreet time orthogonal channel ofmodified multiple level overlay;

FIG. 52 illustrates the PSDs of multiple layer overlay, modifiedmultiple layer overlay and square root raised cosine;

FIG. 53 illustrates a bandwidth comparison based on −40 dBc out of bandpower bandwidth between multiple layer overlay and square root raisedcosine;

FIG. 54 illustrates equivalent discrete time parallel orthogonalchannels of modified multiple layer overlay;

FIG. 55 illustrates four MLO symbols that are included in a singleblock;

FIG. 56 illustrates the channel power gain of the parallel orthogonalchannels of modified multiple layer overlay with three layers andT_(sym)=3;

FIG. 57 illustrates a spectral efficiency comparison based on ACLR1between modified multiple layer overlay and square root raised cosine;

FIG. 58 illustrates a spectral efficiency comparison between modifiedmultiple layer overlay and square root raised cosine based on OBP;

FIG. 59 illustrates a spectral efficiency comparison based on ACLR1between modified multiple layer overlay and square root raised cosine;

FIG. 60 illustrates a spectral efficiency comparison based on OBPbetween modified multiple layer overlay and square root raised cosine;

FIG. 61 illustrates a block diagram of a baseband transmitter for a lowpass equivalent modified multiple layer overlay system;

FIG. 62 illustrates a block diagram of a baseband receiver for a lowpass equivalent modified multiple layer overlay system;

FIG. 63 illustrates a channel simulator;

FIG. 64 illustrates the generation of bit streams for a QAM modulator;

FIG. 65 illustrates a block diagram of a receiver;

FIG. 66 is a flow diagram illustrating an adaptive QLO process;

FIG. 67 is a flow diagram illustrating an adaptive MDM process;

FIG. 68 is a flow diagram illustrating an adaptive QLO and MDM process

FIG. 69 is a flow diagram illustrating an adaptive QLO and QAM process;

FIG. 70 is a flow diagram illustrating an adaptive QLO, MDM and QAMprocess;

FIG. 71 illustrates the use of a pilot signal to improve channelimpairments;

FIG. 72 is a flowchart illustrating the use of a pilot signal to improvechannel impairment;

FIG. 73 illustrates a channel response and the effects of amplifiernonlinearities;

FIG. 74 illustrates the use of QLO in forward and backward channelestimation processes;

FIG. 75 illustrates the manner in which Hermite Gaussian beams andLaguerre Gaussian beams diverge when transmitted from phased arrayantennas;

FIG. 76A illustrates beam divergence between a transmitting aperture anda receiving aperture;

FIG. 76B illustrates the use of a pair of lenses for reducing beamdivergence;

FIG. 77 illustrates the configuration of an optical fiber communicationsystem;

FIG. 78A illustrates a single mode fiber;

FIG. 78B illustrates multi-core fibers;

FIG. 78C illustrates multi-mode fibers;

FIG. 78D illustrates a hollow core fiber;

FIG. 79 illustrates the first six modes within a step index fiber;

FIG. 80 illustrates the classes of random perturbations within a fiber;

FIG. 81 illustrates the intensity patterns of first order groups withina vortex fiber;

FIGS. 82A and 82B illustrate index separation in first order modes of amulti-mode fiber;

FIG. 83 illustrates a few mode fiber providing a linearly polarized OAMbeam;

FIG. 84 illustrates the transmission of four OAM beams over a fiber;

FIG. 85A illustrates the recovered constellations of 20 Gbit/sec QPSKsignals carried on each OAM beam of the device of FIG. 84;

FIG. 85B illustrates the measured BER curves of the device of FIG. 84;

FIG. 86 illustrates a vortex fiber;

FIG. 87 illustrates intensity profiles and interferograms of OAM beams;

FIG. 88 illustrates a free-space communication system;

FIG. 89 illustrates a block diagram of a free-space optics system usingorbital angular momentum and multi-level overlay modulation;

FIGS. 90A-90C illustrate the manner for multiplexing multiple datachannels into optical links to achieve higher data capacity;

FIG. 90D illustrates groups of concentric rings for a wavelength havingmultiple OAM valves;

FIG. 91 illustrates a WDM channel containing many orthogonal OAM beams;

FIG. 92 illustrates a node of a free-space optical system;

FIG. 93 illustrates a network of nodes within a free-space opticalsystem;

FIG. 94 illustrates a system for multiplexing between a free spacesignal and an RF signal;

FIG. 95 illustrates a seven dimensional QKD link based on OAM encoding;

FIG. 96 illustrates the OAM and ANG modes providing complementary 7dimensional bases for information encoding;

FIG. 97 illustrates a block diagram of an OAM processing systemutilizing quantum key distribution;

FIG. 98 illustrates a basic quantum key distribution system;

FIG. 99 illustrates the manner in which two separate states are combinedinto a single conjugate pair within quantum key distribution;

FIG. 100 illustrates one manner in which 0 and 1 bits may be transmittedusing different basis within a quantum key distribution system;

FIG. 101 is a flow diagram illustrating the process for a transmittertransmitting a quantum key;

FIG. 102 illustrates the manner in which the receiver may receive anddetermine a shared quantum key;

FIG. 103 more particularly illustrates the manner in which a transmitterand receiver may determine a shared quantum key;

FIG. 104 is a flow diagram illustrating the process for determiningwhether to keep or abort a determined key;

FIG. 105 illustrates a functional block diagram of a transmitter andreceiver utilizing a free-space quantum key distribution system;

FIG. 106 illustrates a network cloud-based quantum key distributionsystem;

FIG. 107 illustrates a high-speed single photon detector incommunication with a plurality of users; and

FIG. 108 illustrates a nodal quantum key distribution network.

FIG. 109 illustrates the use of a reflective phase hologram for dataexchange;

FIG. 110 is a flow diagram illustrating the process for using ROADM forexchanging data signals;

FIG. 111 illustrates the concept of a ROADM for data channels carried onmultiplexed OAM beams;

FIG. 112 illustrates observed intensity profiles at each step of anad/drop operation such as that of FIG. 111;

FIG. 113 illustrates circuitry for the generation of an OAM twisted beamusing a hologram within a micro-electromechanical device;

FIG. 114 illustrates multiple holograms generated by amicro-electromechanical device;

FIG. 115 illustrates a square array of holograms on a dark background;

FIG. 116 illustrates a hexagonal array of holograms on a darkbackground;

FIG. 117 illustrates a process for multiplexing various OAM modestogether;

FIG. 118 illustrates fractional binary fork holograms;

FIG. 119 illustrates an array of square holograms with no separation ona light background and associated generated OAM mode image;

FIG. 120 illustrates an array of circular holograms separated on a lightbackground and associated generated OAM mode image;

FIG. 121 illustrates an array of square holograms with no separation ona dark background and associated generated OAM mode image;

FIG. 122 illustrates an array of circular holograms on a dark backgroundand associated generated OAM mode image;

FIG. 123 illustrates circular holograms with separation on a brightbackground and associated generated OAM mode image;

FIG. 124 illustrates circular holograms with separation on a darkbackground and associated generated OAM mode image;

FIG. 125 illustrates a hexagonal array of circular holograms on a brightbackground and associated OAM mode image;

FIG. 126 illustrates an hexagonal array of small holograms on a brightbackground and associated OAM mode image;

FIG. 127 illustrates a hexagonal array of circular holograms on a darkbackground and associated OAM mode image;

FIG. 128 illustrates a hexagonal array of small holograms on a darkbackground and associated OAM mode image;

FIG. 129 illustrates a hexagonal array of small holograms separated on adark background and associated OAM mode image;

FIG. 130 illustrates a hexagonal array of small holograms closelylocated on a dark background and associated OAM mode image;

FIG. 131 illustrates a hexagonal array of small holograms that areseparated on a bright background and associated OAM mode image;

FIG. 132 illustrates a hexagonal array of small holograms that areclosely located on a bright background and associated OAM mode image;

FIG. 133 illustrates reduced binary holograms having a radius equal to100 micro-mirrors and a period of 50 for various OAM modes;

FIG. 134 illustrates OAM modes for holograms having a radius of 50micro-mirrors and a period of 50;

FIG. 135 illustrates OAM modes for holograms having a radius of 100micro-mirrors and a period of 100;

FIG. 136 illustrates OAM modes for holograms having a radius of 50micro-mirrors and a period of 50;

FIG. 137 illustrates additional methods of multimode OAM generation byimplementing multiple holograms within a MEMs device;

FIG. 138 illustrates binary spiral holograms;

FIG. 139 is a block diagram of a circuit for generating a muxed andmultiplexed data stream containing multiple new Eigen channels;

FIG. 140 is a flow diagram describing the operation of the circuit ofFIG. 139;

FIG. 141 is a block diagram of a circuit for de-muxing andde-multiplexing a data stream containing multiple new Eigen channels;

FIG. 142 is a flow diagram describing the operation of the circuit ofFIG. 141;

FIG. 143 illustrates a single input, single output (SISO) channel;

FIG. 144 illustrates a multiple input, multiple output (MIMO) channel;

FIG. 145 illustrates the manner in which a MIMO channel increasescapacity without increasing power;

FIG. 146 compares capacity between a MIMO system and a single channelsystem;

FIG. 147 illustrates multiple links provided by a MIMO system;

FIG. 148 illustrates various types of channels between a transmitter anda receiver;

FIG. 149 illustrates an SISO system, MIMO diversity system and the MIMOmultiplexing system;

FIG. 150 illustrates the loss coefficients of a 2×2 MIMO channel overtime;

FIG. 151 illustrates the manner in which the bit error rate declines asa function of the exponent of the signal-to-noise ratio;

FIG. 152 illustrates diversity gains in a fading channel;

FIG. 153 illustrates a model decomposition of a MIMO channel with fullCSI;

FIG. 154 illustrates SVD decomposition of a matrix channel into parallelequivalent channels;

FIG. 155 illustrates a system channel model;

FIG. 156 illustrates the receive antenna distance versus correlation;

FIG. 157 illustrates the manner in which correlation reduces capacity infrequency selective channels;

FIG. 158 illustrates the manner in which channel information varies withfrequency in a frequency selective channel;

FIG. 159 illustrates antenna placement in a MIMO system;

FIG. 160 illustrates multiple communication links at a MIMO receiver;

FIG. 161 illustrates the increased bandwidth provided by MIMO techniquesand mode division multiplexing (MDM) techniques;

FIG. 162 illustrates a combined MDM and MIMO transmitter/receiversystem;

FIG. 163 illustrates a combined MDM, MRC and MIMO transmitter/receiversystem; and

FIG. 164 illustrates a combined MRC and MIMO transmitter/receiversystem.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are usedherein to designate like elements throughout, the various views andembodiments of a system and method for combining MIMO and mode-divisionmultiplexing (MDM) techniques for communications are illustrated anddescribed, and other possible embodiments are described. The figures arenot necessarily drawn to scale, and in some instances the drawings havebeen exaggerated and/or simplified in places for illustrative purposesonly. One of ordinary skill in the art will appreciate the many possibleapplications and variations based on the following examples of possibleembodiments.

Achieving higher data capacity is perhaps one of the primary interest ofthe communications community. This is led to the investigation of usingdifferent physical properties of a light wave for communications,including amplitude, phase, wavelength and polarization. Orthogonalmodes in spatial positions are also under investigation and seemed to beuseful as well. Generally these investigative efforts can be summarizedin 2 categories: 1) encoding and decoding more bets on a single opticalpulse; a typical example is the use of advanced modulation formats,which encode information on amplitude, phase and polarization states,and 2) multiplexing and demultiplexing technologies that allow parallelpropagation of multiple independent data channels, each of which isaddressed by different light property (e.g., wavelength, polarizationand space, corresponding to wavelength-division multiplexing (WDM),polarization-division multiplexing (PDM) and space division multiplexing(SDM), respectively).

The recognition that orbital angular momentum (OAM) has applications incommunication has made it an interesting research topic. It iswell-known that a photon can carry both spin angular momentum andorbital angular momentum. Contrary to spin angular momentum (e.g.,circularly polarized light), which is identified by the electrical fielderection, OAM is usually carried by a light beam with a helical phasefront. Due to the helical phase structure, an OAM carrying beam usuallyhas an annular intensity profile with a phase singularity at the beamcenter. Importantly, depending on discrete twisting speed of the helicalphase, OAM beams can be quantified is different states, which arecompletely distinguishable while propagating coaxially. This propertyallows OAM beams to be potentially useful in either of the 2aforementioned categories to help improve the performance of a freespace or fiber communication system. Specifically, OAM states could beused as a different dimension to encode bits on a single pulse (or asingle photon), or be used to create additional data carriers in an SDMsystem.

There are some potential benefits of using OAM for communications, somespecially designed novel fibers allow less mode coupling and cross talkwhile propagating in fibers. In addition, OAM beams with differentstates share a ring-shaped beam profile, which indicate rotationalinsensitivity for receiving the beams. Since the distinction of OAMbeams does not rely on the wavelength or polarization, OAM multiplexingcould be used in addition to WDM and PDM techniques so that potentiallyimprove the system performance may be provided.

Referring now to the drawings, and more particularly to FIG. 1, whereinthere is illustrated two manners for increasing spectral efficiency of acommunications system. In general, there are basically two ways toincrease spectral efficiency 102 of a communications system. Theincrease may be brought about by signal processing techniques 104 in themodulation scheme or using multiple access technique. Additionally, thespectral efficiency can be increase by creating new Eigen channels 106within the electromagnetic propagation. These two techniques arecompletely independent of one another and innovations from one class canbe added to innovations from the second class. Therefore, thecombination of this technique introduced a further innovation.

Spectral efficiency 102 is the key driver of the business model of acommunications system. The spectral efficiency is defined in units ofbit/sec/hz and the higher the spectral efficiency, the better thebusiness model. This is because spectral efficiency can translate to agreater number of users, higher throughput, higher quality or some ofeach within a communications system.

Regarding techniques using signal processing techniques or multipleaccess techniques. These techniques include innovations such as TDMA,FDMA, CDMA, EVDO, GSM, WCDMA, HSPA and the most recent OFDM techniquesused in 4G WIMAX and LTE. Almost all of these techniques use decades-oldmodulation techniques based on sinusoidal Eigen functions called QAMmodulation. Within the second class of techniques involving the creationof new Eigen channels 106, the innovations include diversity techniquesincluding space and polarization diversity as well as multipleinput/multiple output (MIMO) where uncorrelated radio paths createindependent Eigen channels and propagation of electromagnetic waves.

Referring now to FIG. 2, the present communication system configurationintroduces two techniques, one from the signal processing techniques 104category and one from the creation of new eigen channels 106 categorythat are entirely independent from each other. Their combinationprovides a unique manner to disrupt the access part of an end to endcommunications system from twisted pair and cable to fiber optics, tofree space optics, to RF used in cellular, backhaul and satellite, to RFsatellite, to RF broadcast, to RF point-to point, to RFpoint-to-multipoint, to RF point-to-point (backhaul), to RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet), toInternet of Things (TOT), to Wi-Fi, to Bluetooth, to a personal devicecable replacement, to an RF and FSO hybrid system, to Radar, toelectromagnetic tags and to all types of wireless access. The firsttechnique involves the use of a new signal processing technique usingnew orthogonal signals to upgrade QAM modulation using non sinusoidalfunctions. This is referred to as quantum level overlay (QLO) 202. Thesecond technique involves the application of new electromagneticwavefronts using a property of electromagnetic waves or photon, calledorbital angular momentum (QAM) 104. Application of each of the quantumlevel overlay techniques 202 and orbital angular momentum application204 uniquely offers orders of magnitude higher spectral efficiency 206within communication systems in their combination.

With respect to the quantum level overlay technique 202, new eigenfunctions are introduced that when overlapped (on top of one anotherwithin a symbol) significantly increases the spectral efficiency of thesystem. The quantum level overlay technique 302 borrows from quantummechanics, special orthogonal signals that reduce the time bandwidthproduct and thereby increase the spectral efficiency of the channel.Each orthogonal signal is overlaid within the symbol acts as anindependent channel. These independent channels differentiate thetechnique from existing modulation techniques.

With respect to the application of orbital angular momentum 204, thistechnique introduces twisted electromagnetic waves, or light beams,having helical wave fronts that carry orbital angular momentum (OAM).Different OAM carrying waves/beams can be mutually orthogonal to eachother within the spatial domain, allowing the waves/beams to beefficiently multiplexed and demultiplexed within a communications link.OAM beams are interesting in communications due to their potentialability in special multiplexing multiple independent data carryingchannels.

With respect to the combination of quantum level overlay techniques 202and orbital angular momentum application 204, the combination is uniqueas the OAM multiplexing technique is compatible with otherelectromagnetic techniques such as wave length and polarization divisionmultiplexing. This suggests the possibility of further increasing systemperformance. The application of these techniques together in highcapacity data transmission disrupts the access part of an end to endcommunications system from twisted pair and cable to fiber optics, tofree space optics, to RF used in cellular, backhaul and satellite, to RFsatellite, to RF broadcast, to RF point-to point, to RFpoint-to-multipoint, to RF point-to-point (backhaul), to RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet), toInternet of Things (IOT), to Wi-Fi, to Bluetooth, to a personal devicecable replacement, to an RF and FSO hybrid system, to Radar, toelectromagnetic tags and to all types of wireless access.

Each of these techniques can be applied independent of one another, butthe combination provides a unique opportunity to not only increasespectral efficiency, but to increase spectral efficiency withoutsacrificing distance or signal to noise ratios.

Using the Shannon Capacity Equation, a determination may be made ifspectral efficiency is increased. This can be mathematically translatedto more bandwidth. Since bandwidth has a value, one can easily convertspectral efficiency gains to financial gains for the business impact ofusing higher spectral efficiency. Also, when sophisticated forward errorcorrection (FEC) techniques are used, the net impact is higher qualitybut with the sacrifice of some bandwidth. However, if one can achievehigher spectral efficiency (or more virtual bandwidth), one cansacrifice some of the gained bandwidth for FEC and therefore higherspectral efficiency can also translate to higher quality.

Telecom operators and vendors are interested in increasing spectralefficiency. However, the issue with respect to this increase is thecost. Each technique at different layers of the protocol has a differentprice tag associated therewith. Techniques that are implemented at aphysical layer have the most impact as other techniques can besuperimposed on top of the lower layer techniques and thus increase thespectral efficiency further. The price tag for some of the techniquescan be drastic when one considers other associated costs. For example,the multiple input multiple output (MIMO) technique uses additionalantennas to create additional paths where each RF path can be treated asan independent channel and thus increase the aggregate spectralefficiency. In the MIMO scenario, the operator has other associated softcosts dealing with structural issues such as antenna installations, etc.These techniques not only have tremendous cost, but they have hugetiming issues as the structural activities take time and the achievingof higher spectral efficiency comes with significant delays which canalso be translated to financial losses.

The quantum level overlay technique 202 has an advantage that theindependent channels are created within the symbols without needing newantennas. This will have a tremendous cost and time benefit compared toother techniques. Also, the quantum layer overlay technique 202 is aphysical layer technique, which means there are other techniques athigher layers of the protocol that can all ride on top of the QLOtechniques 202 and thus increase the spectral efficiency even further.QLO technique 202 uses standard QAM modulation used in OFDM basedmultiple access technologies such as WIMAX or LTE. QLO technique 202basically enhances the QAM modulation at the transceiver by injectingnew signals to the I & Q components of the baseband and overlaying thembefore QAM modulation as will be more fully described herein below. Atthe receiver, the reverse procedure is used to separate the overlaidsignal and the net effect is a pulse shaping that allows betterlocalization of the spectrum compared to standard QAM or even the rootraised cosine. The impact of this technique is a significantly higherspectral efficiency.

Referring now more particularly to FIG. 3, there is illustrated ageneral overview of the manner for providing improved communicationbandwidth within various communication protocol interfaces 302, using acombination of multiple level overlay modulation 304 and the applicationof orbital angular momentum 306 to increase the number of communicationschannels.

The various communication protocol interfaces 302 may comprise a varietyof communication links, such as RF communication, wireline communicationsuch as cable or twisted pair connections, or optical communicationsmaking use of light wavelengths such as fiber-optic communications orfree-space optics. Various types of RF communications may include acombination of RF microwave or RF satellite communication, as well asmultiplexing between RF and free-space optics in real time.

By combining a multiple layer overlay modulation technique 304 withorbital angular momentum (OAM) technique 306, a higher throughput overvarious types of communication links 302 may be achieved. The use ofmultiple level overlay modulation alone without OAM increases thespectral efficiency of communication links 302, whether wired, optical,or wireless. However, with OAM, the increase in spectral efficiency iseven more significant.

Multiple overlay modulation techniques 304 provide a new degree offreedom beyond the conventional 2 degrees of freedom, with time T andfrequency F being independent variables in a two-dimensional notationalspace defining orthogonal axes in an information diagram. This comprisesa more general approach rather than modeling signals as fixed in eitherthe frequency or time domain. Previous modeling methods using fixed timeor fixed frequency are considered to be more limiting cases of thegeneral approach of using multiple level overlay modulation 304. Withinthe multiple level overlay modulation technique 304, signals may bedifferentiated in two-dimensional space rather than along a single axis.Thus, the information-carrying capacity of a communications channel maybe determined by a number of signals which occupy different time andfrequency coordinates and may be differentiated in a notationaltwo-dimensional space.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity, within an allocated channel.Given the frequency channel delta (Δf), a given signal transmittedthrough it in minimum time Δt will have an envelope described by certaintime-bandwidth minimizing signals. The time-bandwidth products for thesesignals take the form:

ΔtΔf=½(2n+1)

where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms.

The orbital angular momentum process 306 provides a twist to wave frontsof the electromagnetic fields carrying the data stream that may enablethe transmission of multiple data streams on the same frequency,wavelength, or other signal-supporting mechanism. Similarly, otherorthogonal signals may be applied to the different data streams toenable transmission of multiple data streams on the same frequency,wavelength or other signal-supporting mechanism. This will increase thebandwidth over a communications link by allowing a single frequency orwavelength to support multiple eigen channels, each of the individualchannels having a different orthogonal and independent orbital angularmomentum associated therewith.

Referring now to FIG. 4, there is illustrated a further communicationimplementation technique using the above described techniques as twistedpairs or cables carry electrons (not photons). Rather than using each ofthe multiple level overlay modulation 304 and orbital angular momentumtechniques 306, only the multiple level overlay modulation 304 can beused in conjunction with a single wireline interface and, moreparticularly, a twisted pair communication link or a cable communicationlink 402. The operation of the multiple level overlay modulation 404, issimilar to that discussed previously with respect to FIG. 3, but is usedby itself without the use of orbital angular momentum techniques 306,and is used with either a twisted pair communication link or cableinterface communication link 402 or with fiber optics, free spaceoptics, RF used in cellular, backhaul and satellite, RF satellite, RFbroadcast, RF point-to point, RF point-to-multipoint, RF point-to-point(backhaul), RF point-to-point (fronthaul to provide higher throughputCPRI interface for cloudification and virtualization of RAN andcloudified HetNet), Internet of Things (IOT), Wi-Fi, Bluetooth, apersonal device cable replacement, an RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access.

Referring now to FIG. 5, there is illustrated a general block diagramfor processing a plurality of data streams 502 for transmission in anoptical communication system. The multiple data streams 502 are providedto the multi-layer overlay modulation circuitry 504 wherein the signalsare modulated using the multi-layer overlay modulation technique. Themodulated signals are provided to orbital angular momentum processingcircuitry 506 which applies a twist to each of the wave fronts beingtransmitted on the wavelengths of the optical communication channel. Thetwisted waves are transmitted through the optical interface 508 over anoptical or other communications link such as an optical fiber or freespace optics communication system. FIG. 5 may also illustrate an RFmechanism wherein the interface 508 would comprise and RF interfacerather than an optical interface.

Referring now more particularly to FIG. 6, there is illustrated afunctional block diagram of a system for generating the orbital angularmomentum “twist” within a communication system, such as that illustratedwith respect to FIG. 3, to provide a data stream that may be combinedwith multiple other data streams for transmission upon a same wavelengthor frequency. Multiple data streams 602 are provided to the transmissionprocessing circuitry 600. Each of the data streams 602 comprises, forexample, an end to end link connection carrying a voice call or a packetconnection transmitting non-circuit switch packed data over a dataconnection. The multiple data streams 602 are processed bymodulator/demodulator circuitry 604. The modulator/demodulator circuitry604 modulates the received data stream 602 onto a wavelength orfrequency channel using a multiple level overlay modulation technique,as will be more fully described herein below. The communications linkmay comprise an optical fiber link, free-space optics link, RF microwavelink, RF satellite link, wired link (without the twist), etc.

The modulated data stream is provided to the orbital angular momentum(OAM) signal processing block 606. The orbital angular momentum signalprocessing block 606 applies in one embodiment an orbital angularmomentum to a signal. In other embodiments the processing block 606 canapply any orthogonal function to a signal being transmitted. Theseorthogonal functions can be spatial Bessel functions, Laguerre-Gaussianfunctions, Hermite-Gaussian functions, Ince-Gaussian functions or anyother orthogonal function. Each of the modulated data streams from themodulator/demodulator 604 are provided a different orbital angularmomentum by the orbital angular momentum electromagnetic block 606 suchthat each of the modulated data streams have a unique and differentorbital angular momentum associated therewith. Each of the modulatedsignals having an associated orbital angular momentum are provided to anoptical transmitter 608 that transmits each of the modulated datastreams having a unique orbital angular momentum on a same wavelength.Each wavelength has a selected number of bandwidth slots B and may haveits data transmission capability increase by a factor of the number ofdegrees of orbital angular momentum 1 that are provided from the OAMelectromagnetic block 606. The optical transmitter 608 transmittingsignals at a single wavelength could transmit B groups of information.The optical transmitter 608 and OAM electromagnetic block 606 maytransmit 1× B groups of information according to the configurationdescribed herein.

In a receiving mode, the optical transmitter 608 will have a wavelengthincluding multiple signals transmitted therein having different orbitalangular momentum signals embedded therein. The optical transmitter 608forwards these signals to the OAM signal processing block 606, whichseparates each of the signals having different orbital angular momentumand provides the separated signals to the demodulator circuitry 604. Thedemodulation process extracts the data streams 602 from the modulatedsignals and provides it at the receiving end using the multiple layeroverlay demodulation technique.

Referring now to FIG. 7, there is provided a more detailed functionaldescription of the OAM signal processing block 606. Each of the inputdata streams are provided to OAM circuitry 702. Each of the OAMcircuitry 702 provides a different orbital angular momentum to thereceived data stream. The different orbital angular momentums areachieved by applying different currents for the generation of thesignals that are being transmitted to create a particular orbitalangular momentum associated therewith. The orbital angular momentumprovided by each of the OAM circuitries 702 are unique to the datastream that is provided thereto. An infinite number of orbital angularmomentums may be applied to different input data streams using manydifferent currents. Each of the separately generated data streams areprovided to a signal combiner 704, which combines/multiplexes thesignals onto a wavelength for transmission from the transmitter 706. Thecombiner 704 performs a spatial mode division multiplexing to place allof the signals upon a same carrier signal in the space domain.

Referring now to FIG. 8, there is illustrated the manner in which theOAM processing circuitry 606 may separate a received signal intomultiple data streams. The receiver 802 receives the combined OAMsignals on a single wavelength and provides this information to a signalseparator 804. The signal separator 804 separates each of the signalshaving different orbital angular momentums from the received wavelengthand provides the separated signals to OAM de-twisting circuitry 806. TheOAM de-twisting circuitry 806 removes the associated OAM twist from eachof the associated signals and provides the received modulated datastream for further processing. The signal separator 804 separates eachof the received signals that have had the orbital angular momentumremoved therefrom into individual received signals. The individuallyreceived signals are provided to the receiver 802 for demodulationusing, for example, multiple level overlay demodulation as will be morefully described herein below.

FIG. 9 illustrates in a manner in which a single wavelength orfrequency, having two quanti-spin polarizations may provide an infinitenumber of twists having various orbital angular momentums associatedtherewith. The l axis represents the various quantized orbital angularmomentum states which may be applied to a particular signal at aselected frequency or wavelength. The symbol omega (ω) represents thevarious frequencies to which the signals of differing orbital angularmomentum may be applied. The top grid 902 represents the potentiallyavailable signals for a left handed signal polarization, while thebottom grid 904 is for potentially available signals having right handedpolarization.

By applying different orbital angular momentum states to a signal at aparticular frequency or wavelength, a potentially infinite number ofstates may be provided at the frequency or wavelength. Thus, the stateat the frequency Δω or wavelength 906 in both the left handedpolarization plane 902 and the right handed polarization plane 904 canprovide an infinite number of signals at different orbital angularmomentum states Δl. Blocks 908 and 910 represent a particular signalhaving an orbital angular momentum Δl at a frequency Δω or wavelength inboth the right handed polarization plane 904 and left handedpolarization plane 910, respectively. By changing to a different orbitalangular momentum within the same frequency Δω or wavelength 906,different signals may also be transmitted. Each angular momentum statecorresponds to a different determined current level for transmissionfrom the optical transmitter. By estimating the equivalent current forgenerating a particular orbital angular momentum within the opticaldomain and applying this current for transmission of the signals, thetransmission of the signal may be achieved at a desired orbital angularmomentum state.

Thus, the illustration of FIG. 9, illustrates two possible angularmomentums, the spin angular momentum, and the orbital angular momentum.The spin version is manifested within the polarizations of macroscopicelectromagnetism, and has only left and right hand polarizations due toup and down spin directions. However, the orbital angular momentumindicates an infinite number of states that are quantized. The paths aremore than two and can theoretically be infinite through the quantizedorbital angular momentum levels.

It is well-known that the concept of linear momentum is usuallyassociated with objects moving in a straight line. The object could alsocarry angular momentum if it has a rotational motion, such as spinning(i.e., spin angular momentum (SAM) 1002), or orbiting around an axis1006 (i.e., OAM 1004), as shown in FIGS. 10A and 10B, respectively. Alight beam may also have rotational motion as it propagates. In paraxialapproximation, a light beam carries SAM 1002 if the electrical fieldrotates along the beam axis 1006 (i.e., circularly polarized light1005), and carries OAM 1004 if the wave vector spirals around the beamaxis 1006, leading to a helical phase front 1008, as shown in FIGS. 10Cand 10D. In its analytical expression, this helical phase front 1008 isusually related to a phase term of exp(il θ) in the transverse plane,where θ refers to the angular coordinate, and l is an integer indicatingthe number of intertwined helices (i.e., the number of 2π phase shiftsalong the circle around the beam axis). l could be a positive, negativeinteger or zero, corresponding to clockwise, counterclockwise phasehelices or a Gaussian beam with no helix, respectively.

Two important concepts relating to OAM include: 1) OAM and polarization:As mentioned above, an OAM beam is manifested as a beam with a helicalphase front and therefore a twisting wavevector, while polarizationstates can only be connected to SAM 1002. A light beam carries SAM 1002of ±h/2π (h is Plank's constant) per photon if it is left or rightcircularly polarized, and carries no SAM 1002 if it is linearlypolarized. Although the SAM 1002 and OAM 1004 of light can be coupled toeach other under certain scenarios, they can be clearly distinguishedfor a paraxial light beam. Therefore, with the paraxial assumption, OAM1004 and polarization can be considered as two independent properties oflight.

2) OAM beam and Laguerre-Gaussian (LG) beam: In general, an OAM-carryingbeam could refer to any helically phased light beam, irrespective of itsradial distribution (although sometimes OAM could also be carried by anon-helically phased beam). LG beam is a special subset among allOAM-carrying beams, due to that the analytical expression of LG beamsare eigen-solutions of paraxial form of the wave equation in acylindrical coordinates. For an LG beam, both azimuthal and radialwavefront distributions are well defined, and are indicated by two indexnumbers, l and p, of which l has the same meaning as that of a generalOAM beam, and p refers to the radial nodes in the intensitydistribution. Mathematical expressions of LG beams form an orthogonaland complete basis in the spatial domain. In contrast, a general OAMbeam actually comprises a group of LG beams (each with the same l indexbut a different p index) due to the absence of radial definition. Theterm of “OAM beam” refers to all helically phased beams, and is used todistinguish from LG beams.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the radiationtransmitted by the signals. The Maxwell-Heaviside equations can berepresented as:

${\nabla{\cdot E}} = \frac{\rho}{ɛ_{0}}$${\nabla{\times E}} = {- \frac{\partial B}{\partial t}}$ ∇⋅B = 0${\nabla{\times B}} = {{ɛ_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,x} \right)}}}$

where ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, one can derive 23symmetries/conserved quantities from Maxwell's original equations.However, there are only ten well-known conserved quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

Maxwell's linear theory is of U(1) symmetry with Abelian commutationrelations. They can be extended to higher symmetry group SU(2) form withnon-Abelian commutation relations that address global (non-local inspace) properties. The Wu-Yang and Harmuth interpretation of Maxwell'stheory implicates the existence of magnetic monopoles and magneticcharges. As far as the classical fields are concerned, these theoreticalconstructs are pseudo-particle, or instanton. The interpretation ofMaxwell's work actually departs in a significant ways from Maxwell'soriginal intention. In Maxwell's original formulation, Faraday'selectronic states (the Aμ field) was central making them compatible withYang-Mills theory (prior to Heaviside). The mathematical dynamicentities called solutions can be either classical or quantum, linear ornon-linear and describe EM waves. However, solutions are of SU(2)symmetry forms. In order for conventional interpreted classicalMaxwell's theory of U(1) symmetry to describe such entities, the theorymust be extended to SU(2) forms.

Besides the half dozen physical phenomena (that cannot be explained withconventional Maxwell's theory), the recently formulated Harmuth Ansatzalso address the incompleteness of Maxwell's theory. Harmuth amendedMaxwell's equations can be used to calculate EM signal velocitiesprovided that a magnetic current density and magnetic charge are addedwhich is consistent to Yang-Mills filed equations. Therefore, with thecorrect geometry and topology, the Aμ potentials always have physicalmeaning

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

$H = {{\sum\limits_{i}{m_{i}\gamma_{i}c^{2}}} + {\frac{ɛ_{0}}{2}{\int{d^{3}{x\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}}$Hamiltonian  (total  energy)${\frac{{dU}^{mech}}{dt} + \frac{{dU}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\hat{n^{\prime}} \cdot S}}}} = 0$conservation  of  energy

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

${p = {{\sum\limits_{i}{m_{i}\gamma_{i}v_{i}}} + {ɛ_{0}{\int{d^{3}{x\left( {E \times B} \right)}}}}}}\;$linear  momentum${{\frac{{dp}^{mech}}{dt} + \frac{{dp}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\hat{n^{\prime}} \cdot T}}}} = 0}\;$conservation  of  linear  momentum

The conservation of system center of energy is represented by theequation:

$R = {{\frac{1}{H}{\sum\limits_{i}{\left( {x_{i} - x_{0}} \right)m_{i}\gamma_{i}c^{2}}}} + {\frac{ɛ_{0}}{2H}{\int{d^{3}{x\left( {x - x_{0}} \right)}\left( {{E^{2}} + {c^{2}{B^{2}}}} \right)}}}}$

Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

${{\frac{{dJ}^{mech}}{dt} + \frac{{dJ}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\hat{n^{\prime}} \cdot M}}}} = 0}\;$conservation  of  angular  momentum

For radiation beams in free space, the EM field angular momentum J^(em)can be separated into two parts:

J ^(em)=ε₀∫_(V′) d ³ x′(E×A)+ε₀∫_(V′) d ³ x′E _(i)[(x′−x ₀)×∇]A _(i)

For each singular Fourier mode in real valued representation:

$J^{em} = {{{- i}\; \frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {i\; \frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}$

The first part is the EM spin angular momentum S^(em), its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum L^(em) its classical manifestation is wavehelicity. In general, both EM linear momentum P^(em), and EM angularmomentum J^(em)=L^(em)+S^(em) are radiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

${{\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0},{{continuity}\mspace{14mu} {equation}}$

where S is the Poynting vector

${S = {\frac{1}{4}\left( {{E \times H^{*}} + {E^{*} \times H}} \right)}},$

and U is the energy density

${U = {\frac{1}{4}\left( {{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}},$

with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$V = {{\nabla{\times v_{opt}}} = {\nabla{\times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}}}$

Referring now to FIGS. 11A and 11B, there is illustrated the manner inwhich a signal and its associated Poynting vector in a plane wavesituation. In the plane wave situation illustrated generally at 1002,the transmitted signal may take one of three configurations. When theelectric field vectors are in the same direction, a linear signal isprovided, as illustrated generally at 1004. Within a circularpolarization 1006, the electric field vectors rotate with the samemagnitude. Within the elliptical polarization 1008, the electric fieldvectors rotate but have differing magnitudes. The Poynting vectorremains in a constant direction for the signal configuration to FIG. 10Aand always perpendicular to the electric and magnetic fields. Referringnow to FIG. 10B, when a unique orbital angular momentum is applied to asignal as described here and above, the Poynting vector S 1010 willspiral about the direction of propagation of the signal. This spiral maybe varied in order to enable signals to be transmitted on the samefrequency as described herein.

FIGS. 12A through 12C illustrate the differences in signals havingdifferent helicity (i.e., orbital angular momentums). Each of thespiraling Poynting vectors associated with the signals 1102, 1104, and1106 provide a different shaped signal. Signal 1102 has an orbitalangular momentum of +1, signal 1104 has an orbital angular momentum of+3, and signal 1106 has an orbital angular momentum of −4. Each signalhas a distinct angular momentum and associated Poynting vector enablingthe signal to be distinguished from other signals within a samefrequency. This allows differing type of information to be transmittedon the same frequency, since these signals are separately detectable anddo not interfere with each other (Eigen channels).

FIG. 12 illustrates the propagation of Poynting vectors for variousEigen modes. Each of the rings 1120 represents a different Eigen mode ortwist representing a different orbital angular momentum within the samefrequency. Each of these rings 1120 represents a different orthogonalchannel. Each of the Eigen modes has a Poynting vector 1122 associatedtherewith.

Topological charge may be multiplexed to the frequency for either linearor circular polarization. In case of linear polarizations, topologicalcharge would be multiplexed on vertical and horizontal polarization. Incase of circular polarization, topological charge would multiplex onleft hand and right hand circular polarizations. The topological chargeis another name for the helicity index “I” or the amount of twist or OAMapplied to the signal. Also, use of the orthogonal functions discussedherein above may also be multiplexed together onto a same signal inorder to transmit multiple streams of information. The helicity indexmay be positive or negative. In wireless communications, differenttopological charges/orthogonal functions can be created and muxedtogether and de-muxed to separate the topological chargescharges/orthogonal functions. The signals having different orthogonalfunction are spatially combined together on a same signal but do notinterfere with each other since they are orthogonal to each other.

The topological charges 1 s can be created using Spiral Phase Plates(SPPs) as shown in FIG. 11E using a proper material with specific indexof refraction and ability to machine shop or phase mask, hologramscreated of new materials or a new technique to create an RF version ofSpatial Light Modulator (SLM) that does the twist of the RF waves (asopposed to optical beams) by adjusting voltages on the device resultingin twisting of the RF waves with a specific topological charge. SpiralPhase plates can transform a RF plane wave (1=0) to a twisted RF wave ofa specific helicity (i.e. 1=+1).

Cross talk and multipath interference can be corrected using RFMultiple-Input-Multiple-Output (MIMO). Most of the channel impairmentscan be detected using a control or pilot channel and be corrected usingalgorithmic techniques (closed loop control system).

While the application of orbital angular momentum to various signalsallow the signals to be orthogonal to each other and used on a samesignal carrying medium, other orthogonal function/signals can be appliedto data streams to create the orthogonal signals on the same signalmedia carrier.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity, within an allocated channel.Given the frequency channel delta (Δf), a given signal transmittedthrough it in minimum time Δt will have an envelope described by certaintime-bandwidth minimizing signals. The time-bandwidth products for thesesignals take the form;

ΔtΔf=½(2n+1)

where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms. These types of orthogonal signals that reduce the timebandwidth product and thereby increase the spectral efficiency of thechannel.

Hermite-Gaussian polynomials are one example of a classical orthogonalpolynomial sequence, which are the Eigenstates of a quantum harmonicoscillator. Signals based on Hermite-Gaussian polynomials possess theminimal time-bandwidth product property described above, and may be usedfor embodiments of MLO systems. However, it should be understood thatother signals may also be used, for example orthogonal polynomials suchas Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials, Laguerre-Gaussian polynomials, Hermite-Gaussianpolynomials and Ince-Gaussian polynomials. Q-functions are another classof functions that can be employed as a basis for MLO signals.

In addition to the time bandwidth minimization described above, theplurality of data streams can be processed to provide minimization ofthe Space-Momentum products in spatial modulation. In this case:

${\Delta \; x\; \Delta \; p} = \frac{1}{2}$

Processing of the data streams in this manner create wavefronts that arespatial. The processing creates wavefronts that are also orthogonal toeach other like the OAM twisted functions but these comprise differenttypes of orthogonal functions that are in the spatial domain rather thanthe temporal domain.

The above described scheme is applicable to twisted pair, coaxial cable,fiber optic, RF satellite, RF broadcast, RF point-to point, RFpoint-to-multipoint, RF point-to-point (backhaul), RF point-to-point(fronthaul to provide higher throughput CPRI interface forcloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE.

Hermite Gaussian Beams

Hermite Gaussian beams may also be used for transmitting orthogonal datastreams. In the scalar field approximation (e.g. neglecting the vectorcharacter of the electromagnetic field), any electric field amplitudedistribution can be represented as a superposition of plane waves, i.e.by:

$E \propto {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}\mspace{14mu} {A\left( {k_{x},k_{y}} \right)}e^{{{ik}_{x}x} + {{ik}_{y}y} + {{ik}_{z}z} + {{iz}\sqrt{k^{2} - k_{x}^{2} - k_{y}^{2}}}}}}}$

This representation is also called angular spectrum of plane waves orplane-wave expansion of the electromagnetic field. Here A(k_(x), k_(y))is the amplitude of the plane wave. This representation is chosen insuch a way that the net energy flux connected with the electromagneticfield is towards the propagation axis z. Every plane wave is connectedwith an energy flow that has direction k. Actual lasers generate aspatially coherent electromagnetic field which has a finite transversalextension and propagates with moderate spreading. That means that thewave amplitude changes only slowly along the propagation axis (z-axis)compared to the wavelength and finite width of the beam. Thus, theparaxial approximation can be applied, assuming that the amplitudefunction A(k_(x), k_(y)) falls off sufficiently fast with increasingvalues of (k_(x), k_(y)).

Two principal characteristics of the total energy flux can beconsidered: the divergence (spread of the plane wave amplitudes in wavevector space), defined as:

$\left. {{Divergence} \propto {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}\left( {K_{x}^{2} + K_{y}^{2}} \right)}}}}\mspace{14mu} \middle| {A\left( {k_{x},k_{y}} \right)} \right|^{2}$

and the transversal spatial extension (spread of the field intensityperpendicular to the z-direction) defined as:

$\left. {{{Transversal}\mspace{14mu} {Extension}} \propto {\int_{- \infty}^{\infty}{{dx}{\int_{- \infty}^{\infty}{{dy}\left( {x^{2} + y^{2}} \right)}}}}} \middle| E \right|^{2} = {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}\left\lbrack \left| \frac{\partial A}{\partial x} \middle| {}_{2}{+ \left| \frac{\partial A}{\partial y} \right|^{2}} \right. \right\rbrack}}}$

Let's now look for the fundamental mode of the beam as theelectromagnetic field having simultaneously minimal divergence andminimal transversal extension, i.e. as the field that minimizes theproduct of divergence and extension. By symmetry reasons, this leads tolooking for an amplitude function minimizing the product:

${\left\lbrack \left. {\int_{- \infty}^{\infty}{\frac{{dk}_{x}}{\left( {2\pi} \right)}k_{x}^{2}}} \middle| A \right|^{2} \right\rbrack \left\lbrack \left. {\int_{- \infty}^{\infty}\frac{{dk}_{x}}{\left( {2\pi} \right)}} \middle| \frac{\partial A}{\partial k_{x}} \right|^{2} \right\rbrack} = \frac{\left. ||A \right.||^{4}}{\left( {8\pi^{2}} \right)^{2}}$

Thus, seeking the field with minimal divergence and minimal transversalextension can lead directly to the fundamental Gaussian beam. This meansthat the Gaussian beam is the mode with minimum uncertainty, i.e. theproduct of its sizes in real space and wave-vector space is thetheoretical minimum as given by the Heisenberg's uncertainty principleof Quantum Mechanics. Consequently, the Gaussian mode has lessdispersion than any other optical field of the same size, and itsdiffraction sets a lower threshold for the diffraction of real opticalbeams.

Hermite-Gaussian beams are a family of structurally stable laser modeswhich have rectangular symmetry along the propagation axis. In order toderive such modes, the simplest approach is to include an additionalmodulation of the form:

$E_{m,n}^{H} = {\int_{- \infty}^{\infty}{\frac{{dk}_{x}{dk}_{y}}{\left( {2\pi} \right)^{2}}\left( {ik}_{x} \right)^{m}\mspace{14mu} \left( {ik}_{y} \right)^{n}e^{S}}}$${S\left( {k_{x},k_{y},x,y,z} \right)} = {{{ik}_{x}x} + {{ik}_{y}y} + {{ik}_{z}z} - {\frac{W_{0}}{4}\mspace{14mu} {\left( {1 + {i\frac{Z}{Z_{R}}}} \right)\left\lbrack {k_{x}^{2} + k_{y}^{2}} \right\rbrack}}}$

The new field modes occur to be differential derivatives of thefundamental Gaussian mode E₀.

$E_{m,n}^{H} = {\frac{\partial^{m + n}}{{\partial x^{m}}{\partial y^{n}}}E_{0}}$

Looking at the explicit form E0 shows that the differentiations in thelast equation lead to expressions of the form:

$\frac{\partial^{P}}{\partial x^{p}}e^{({{- \alpha}\; x^{2}})}$

with some constant p and α. Using now the definition of Hermits'polynomials,

${H_{p}(x)} = {\left( {- 1} \right)^{p}e^{(x^{2})}\frac{d^{P}}{{dx}^{p}}e^{({{- \alpha}\; x^{2}})}}$

Then the field amplitude becomes

${E_{m,n}^{H}\left( {x,y,z} \right)} = {\sum\limits_{m}{\sum\limits_{n}{C_{mn}E_{0}\frac{w_{0}}{w(z)}{H_{m}\left( {\sqrt{2}\frac{x}{w(z)}} \right)}\mspace{14mu} H_{n}\mspace{14mu} \left( {\sqrt{2}\frac{y}{w(z)}} \right)\mspace{14mu} e^{\frac{- {({x^{2} + y^{2}})}}{{w{(z)}}^{2}}}e^{{- {j{({m + n + 1})}}}\tan^{- 1}z\text{/}z_{R}}\mspace{14mu} e^{\frac{- {({x^{2} + y^{2}})}}{2{R{(z)}}}}}}}$

Where

ρ² = x² + y² $\xi = \frac{z}{z_{R}}$

and Rayleigh length z_(R)

$z_{R} = \frac{\pi \; w_{0}^{2}}{\lambda}$

And beam diameter

w(ξ)=w ₀√{square root over ((1+ξ²))}

In cylindrical coordinates, the filed takes the form:

${E_{l,p}^{L}\left( {\rho,\phi,z} \right)} = {\sum\limits_{l}{\sum\limits_{np}\mspace{14mu} {C_{lp}E_{0}\frac{w_{0}}{w(z)}\left( {\sqrt{2}\frac{\rho}{w(z)}} \right)^{l}\mspace{14mu} L_{p}^{l}\mspace{14mu} \left( {\sqrt{2}\frac{\rho}{w(z)}} \right)\mspace{14mu} e^{\frac{- \rho^{2}}{{w{(z)}}^{2}}}\mspace{14mu} e^{{- {j{({{2p} + l + 1})}}}\tan^{- 1}z\text{/}z_{R}}\mspace{14mu} e^{{jl}\; \phi}\mspace{14mu} e^{\frac{{- {jk}}\; \rho^{2}}{2{R{(z)}}}}}}}$

Where L^(l) _(p) is Laguerre functions.

Mode division multiplexing (MDM) of multiple orthogonal beams increasesthe system capacity and spectral efficiency in optical communicationsystems. For free space systems, multiple beams each on a differentorthogonal mode can be transmitted through a single transmitter andreceiver aperture pair. Moreover, the modal orthogonality of differentbeans enables the efficient multiplexing at the transmitter anddemultiplexing at the receiver.

Different optical modal basis sets exist that exhibit orthogonality. Forexample, orbital angular momentum (OAM) beams that are either LaguerreGaussian (LG or Laguerre Gaussian light modes may be used formultiplexing of multiple orthogonal beams in free space optical and RFtransmission systems. However, there exist other modal groups that alsomay be used for multiplexing that do not contain OAM. Hermite Gaussian(HG) modes are one such modal group. The intensity of an HG_(m,n) beamis shown according to the equation:

${{I\left( {x,y,z} \right)} = {C_{m,n}H_{m}^{2}\mspace{14mu} \left( \frac{\sqrt{2}x}{w(z)} \right)\mspace{14mu} H_{n}^{2}\mspace{14mu} \left( \frac{\sqrt{2}y}{w(z)} \right) \times {\exp \left( {{- \frac{2x^{2}}{{w(z)}^{2}}} - \frac{2y^{2}}{{w(z)}^{2}}} \right)}}},{{w(z)} = {w_{0}\sqrt{1 + \left\lbrack {\lambda \; z\text{/}\pi \; w_{0}^{2}} \right\rbrack}}}$

in which H_(m)(*) and H_(n)(*) are the Hermite polynomials of the mthand nth order. The value w₀ is the beam waist at distance Z=0. Thespatial orthogonality of HG modes with the same beam waist w₀ relies onthe orthogonality of Hermite polynomial in x or y directions.

Referring now to FIG. 13, there is illustrated a system for using theorthogonality of an HG modal group for free space spatial multiplexingin free space. A laser 1302 is provided to a beam splitter 1304. Thebeam splitter 1304 splits the beam into multiple beams that are eachprovided to a modulator 1306 for modulation with a data stream 1308. Themodulated beam is provided to collimators 1310 that provides acollimated light beam to spatial light modulators 1312. Spatial lightmodulators (SLM's) 1312 may be used for transforming input plane wavesinto HG modes of different orders, each mode carrying an independentdata channel. These HG modes are spatially multiplexed using amultiplexer 1314 and coaxially transmitted over a free space link 1316.Δt the receiver 1318 there are several factors that may affect thedemultiplexing of these HG modes, such as receiver aperture size,receiver lateral displacement and receiver angular error. These factorsaffect the performance of the data channel such as signal-to-noise ratioand crosstalk.

With respect to the characteristics of a diverged HG_(m,0) beam (m=0-6),the wavelength is assumed to be 1550 nm and the transmitted power foreach mode is 0 dBm. Higher order HG modes have been shown to have largerbeam sizes. For smaller aperture sizes less power is received for higherorder HG modes due to divergence.

Since the orthogonality of HG modes relies on the optical fielddistribution in the x and y directions, a finite receiver aperture maytruncate the beam. The truncation will destroy the orthogonality andcost crosstalk of the HG channels. When an aperture is smaller, there ishigher crosstalk to the other modes. When a finite receiver is used, ifan HG mode with an even (odd) order is transmitted, it only causes crosstalk to other HG modes with even (odd) numbers. This is explained by thefact that the orthogonality of the odd and even HG modal groups remainswhen the beam is systematically truncated.

Moreover, misalignment of the receiver may cause crosstalk. In oneexample, lateral displacement can be caused when the receiver is notaligned with the beam axis. In another example, angular error may becaused when the receiver is on axis but there is an angle between thereceiver orientation and the beam propagation axis. As the lateraldisplacement increases, less power is received from the transmittedpower mode and more power is leaked to the other modes. There is lesscrosstalk for the modes with larger mode index spacing from thetransmitted mode.

Referring now to FIG. 14, the reference number 1400 generally indicatesan embodiment of a multiple level overlay (MLO) modulation system,although it should be understood that the term MLO and the illustratedsystem 1400 are examples of embodiments. The MLO system may comprise onesuch as that disclosed in U.S. Pat. No. 8,503,546 entitled MultipleLayer Overlay Modulation which is incorporated herein by reference. Inone example, the modulation system 1400 would be implemented within themultiple level overlay modulation box 504 of FIG. 5. System 1400 takesas input an input data stream 1401 from a digital source 1402, which isseparated into three parallel, separate data streams, 1403A-1403C, oflogical 1s and 0s by input stage demultiplexer (DEMUX) 1404. Data stream1401 may represent a data file to be transferred, or an audio or videodata stream. It should be understood that a greater or lesser number ofseparated data streams may be used. In some of the embodiments, each ofthe separated data streams 1403A-1403C has a data rate of 1/N of theoriginal rate, where N is the number of parallel data streams. In theembodiment illustrated in FIG. 14, N is 3.

Each of the separated data streams 1403A-1403C is mapped to a quadratureamplitude modulation (QAM) symbol in an M-QAM constellation, forexample, 16 QAM or 64 QAM, by one of the QAM symbol mappers 1405A-C. TheQAM symbol mappers 1405A-C are coupled to respective outputs of DEMUX1404, and produced parallel in phase (I) 1406A, 1408A, and 1410A andquadrature phase (Q) 1406B, 1408B, and 1410B data streams at discretelevels. For example, in 64 QAM, each I and Q channel uses 8 discretelevels to transmit 3 bits per symbol. Each of the three I and Q pairs,1406A-1406B, 1408A-1408B, and 1410A-1410B, is used to weight the outputof the corresponding pair of function generators 1407A-1407B,1409A-1409B, and 1411A-1411B, which in some embodiments generate signalssuch as the modified Hermite polynomials described above and weightsthem based on the amplitude value of the input symbols. This provides 2Nweighted or modulated signals, each carrying a portion of the dataoriginally from income data stream 1401, and is in place of modulatingeach symbol in the I and Q pairs, 1406A-1406B, 1408A-1408B, and1410A-1410B with a raised cosine filter, as would be done for a priorart QAM system. In the illustrated embodiment, three signals are used,SH0, SH1, and SH2, which correspond to modifications of H0, H1, and H2,respectively, although it should be understood that different signalsmay be used in other embodiments.

While the description relates to the application of QLO modulation toimprove operation of a quadrature amplitude modulation (QAM) system, theapplication of QLO modulation will also improve the spectral efficiencyof other legacy modulation schemes.

The weighted signals are not subcarriers, but rather are sublayers of amodulated carrier, and are combined, superimposed in both frequency andtime, using summers 1412 and 1416, without mutual interference in eachof the I and Q dimensions, due to the signal orthogonality. Summers 1412and 1416 act as signal combiners to produce composite signals 1413 and1417. The weighted orthogonal signals are used for both I and Qchannels, which have been processed equivalently by system 1400, and aresummed before the QAM signal is transmitted. Therefore, although neworthogonal functions are used, some embodiments additionally use QAM fortransmission. Because of the tapering of the signals in the time domain,as will be shown in FIGS. 18A through 18K, the time domain waveform ofthe weighted signals will be confined to the duration of the symbols.Further, because of the tapering of the special signals and frequencydomain, the signal will also be confined to frequency domain, minimizinginterface with signals and adjacent channels.

The composite signals 1413 and 1417 are converted to analogue signals1415 and 1419 using digital to analogue converters 1414 and 1418, andare then used to modulate a carrier signal at the frequency of localoscillator (LO) 1420, using modulator 1421. Modulator 1421 comprisesmixers 1422 and 1424 coupled to DACs 1414 and 1418, respectively. Ninetydegree phase shifter 1423 converts the signals from LO 1420 into a Qcomponent of the carrier signal. The output of mixers 1422 and 1424 aresummed in summer 1425 to produce output signals 1426.

MLO can be used with a variety of transport mediums, such as wire,optical, and wireless, and may be used in conjunction with QAM. This isbecause MLO uses spectral overlay of various signals, rather thanspectral overlap. Bandwidth utilization efficiency may be increased byan order of magnitude, through extensions of available spectralresources into multiple layers. The number of orthogonal signals isincreased from 2, cosine and sine, in the prior art, to a number limitedby the accuracy and jitter limits of generators used to produce theorthogonal polynomials. In this manner, MLO extends each of the I and Qdimensions of QAM to any multiple access techniques such as GSM, codedivision multiple access (CDMA), wide band CDMA (WCDMA), high speeddownlink packet access (HSPDA), evolution-data optimized (EV-DO),orthogonal frequency division multiplexing (OFDM), world-wideinteroperability for microwave access (WIMAX), and long term evolution(LTE) systems. MLO may be further used in conjunction with othermultiple access (MA) schemes such as frequency division duplexing (FDD),time division duplexing (TDD), frequency division multiple access(FDMA), and time division multiple access (TDMA). Overlaying individualorthogonal signals over the same frequency band allows creation of avirtual bandwidth wider than the physical bandwidth, thus adding a newdimension to signal processing. This modulation is applicable to twistedpair, coaxial cable, fiber optic, RF satellite, RF broadcast, RFpoint-to point, RF point-to-multipoint, RF point-to-point (backhaul), RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE.

Referring now back to FIG. 15, an MLO demodulator 1500 is illustrated,although it should be understood that the term MLO and the illustratedsystem 1500 are examples of embodiments. The modulator 1500 takes asinput an MLO signal 1526 which may be similar to output signal 1526 fromsystem 1400. Synchronizer 1527 extracts phase information, which isinput to local oscillator 1520 to maintain coherence so that themodulator 1521 can produce base band to analogue I signal 1515 and Qsignal 1519. The modulator 1521 comprises mixers 1522 and 1524, which,coupled to OL1520 through 90 degree phase shifter 1523. I signal 1515 isinput to each of signal filters 1507A, 1509A, and 1511A, and Q signal1519 is input to each of signal filters 1507B, 1509B, and 1511B. Sincethe orthogonal functions are known, they can be separated usingcorrelation or other techniques to recover the modulated data.Information in each of the I and Q signals 1515 and 1519 can beextracted from the overlapped functions which have been summed withineach of the symbols because the functions are orthogonal in acorrelative sense.

In some embodiments, signal filters 1507A-1507B, 1509A-1509B, and1511A-1511B use locally generated replicas of the polynomials as knownsignals in match filters. The outputs of the match filters are therecovered data bits, for example, equivalence of the QAM symbols1506A-1506B, 1508A-1508B, and 1510A-1510B of system 1500. Signal filters1507A-1507B, 1509A-1509B, and 1511A-1511B produce 2n streams of n, I,and Q signal pairs, which are input into demodulators 1528-1533.Demodulators 1528-1533 integrate the energy in their respective inputsignals to determine the value of the QAM symbol, and hence the logical1s and 0s data bit stream segment represented by the determined symbol.The outputs of the modulators 1528-1533 are then input into multiplexers(MUXs) 1505A-1505C to generate data streams 1503A-1503C. If system 1500is demodulating a signal from system 1400, data streams 1503A-1503Ccorrespond to data streams 1403A-1403C. Data streams 1503A-1503C aremultiplexed by MUX 1504 to generate data output stream 1501. In summary,MLO signals are overlayed (stacked) on top of one another on transmitterand separated on receiver.

MLO may be differentiated from CDMA or OFDM by the manner in whichorthogonality among signals is achieved. MLO signals are mutuallyorthogonal in both time and frequency domains, and can be overlaid inthe same symbol time bandwidth product. Orthogonality is attained by thecorrelation properties, for example, by least sum of squares, of theoverlaid signals. In comparison, CDMA uses orthogonal interleaving ordisplacement of signals in the time domain, whereas OFDM uses orthogonaldisplacement of signals in the frequency domain.

Bandwidth efficiency may be increased for a channel by assigning thesame channel to multiple users. This is feasible if individual userinformation is mapped to special orthogonal functions. CDMA systemsoverlap multiple user information and views time intersymbol orthogonalcode sequences to distinguish individual users, and OFDM assigns uniquesignals to each user, but which are not overlaid, are only orthogonal inthe frequency domain. Neither CDMA nor OFDM increases bandwidthefficiency. CDMA uses more bandwidth than is necessary to transmit datawhen the signal has a low signal to noise ratio (SNR). OFDM spreads dataover many subcarriers to achieve superior performance in multipathradiofrequency environments. OFDM uses a cyclic prefix OFDM to mitigatemultipath effects and a guard time to minimize intersymbol interference(ISI), and each channel is mechanistically made to behave as if thetransmitted waveform is orthogonal. (Sync function for each subcarrierin frequency domain.)

In contrast, MLO uses a set of functions which effectively form analphabet that provides more usable channels in the same bandwidth,thereby enabling high bandwidth efficiency. Some embodiments of MLO donot require the use of cyclic prefixes or guard times, and therefore,outperforms OFDM in spectral efficiency, peak to average power ratio,power consumption, and requires fewer operations per bit. In addition,embodiments of MLO are more tolerant of amplifier nonlinearities thanare CDMA and OFDM systems.

FIG. 16 illustrates an embodiment of an MLO transmitter system 1600,which receives input data stream 1601. System 1600 represents amodulator/controller, which incorporates equivalent functionality ofDEMUX 1604, QAM symbol mappers 1405A-C, function generators 1407A-1407B,1409A-1409B, and 1411A-1411B, and summers 1412 and 1416 of system 1400,shown in FIG. 14. However, it should be understood thatmodulator/controller 1601 may use a greater or lesser quantity ofsignals than the three illustrated in system 1400. Modulator/controller1601 may comprise an application specific integrated circuit (ASIC), afield programmable gate array (FPGA), and/or other components, whetherdiscrete circuit elements or integrated into a single integrated circuit(IC) chip.

Modulator/controller 1601 is coupled to DACs 1604 and 1607,communicating a 10 bit I signal 1602 and a 10 bit Q signal 1605,respectively. In some embodiments, I signal 1602 and Q signal 1605correspond to composite signals 1413 and 1417 of system 1400. It shouldbe understood, however, that the 10 bit capacity of I signal 1602 and Qsignal 1605 is merely representative of an embodiment. As illustrated,modulator/controller 1601 also controls DACs 1604 and 1607 using controlsignals 1603 and 1606, respectively. In some embodiments, DACs 1604 and1607 each comprise an AD5433, complementary metal oxide semiconductor(CMOS) 10 bit current output DAC. In some embodiments, multiple controlsignals are sent to each of DACs 1604 and 1607.

DACs 1604 and 1607 output analogue signals 1415 and 1419 to quadraturemodulator 1421, which is coupled to LO 1420. The output of modulator1420 is illustrated as coupled to a transmitter 1608 to transmit datawirelessly, although in some embodiments, modulator 1421 may be coupledto a fiber-optic modem, a twisted pair, a coaxial cable, or othersuitable transmission media.

FIG. 15 illustrates an embodiment of an MLO receiver system 1500 capableof receiving and demodulating signals from system 1600. System 1500receives an input signal from a receiver 1508 that may comprise inputmedium, such as RF, wired or optical. The modulator 1321 driven by LO1320 converts the input to baseband I signal 1315 and Q signal 1319. Isignal 1315 and Q signal 1319 are input to analogue to digital converter(ADC) 1509.

ADC 1709 outputs 10 bit signal 1710 to demodulator/controller 1701 andreceives a control signal 1712 from demodulator/controller 1701.Demodulator/controller 1701 may comprise an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA),and/or other components, whether discrete circuit elements or integratedinto a single integrated circuit (IC) chip. Demodulator/controller 1701correlates received signals with locally generated replicas of thesignal set used, in order to perform demodulation and identify thesymbols sent. Demodulator/controller 1701 also estimates frequencyerrors and recovers the data clock, which is used to read data from theADC 1709. The clock timing is sent back to ADC 1709 using control signal1712, enabling ADC 1709 to segment the digital I and Q signals 1517 and1519. In some embodiments, multiple control signals are sent bydemodulator/controller 1701 to ADC 1709. Demodulator/controller 1701also outputs data signal 1301.

Hermite-Gaussian polynomials are a classical orthogonal polynomialsequence, which are the Eigenstates of a quantum harmonic oscillator.Signals based on Hermite-Gaussian polynomials possess the minimaltime-bandwidth product property described above, and may be used forembodiments of MLO systems. However, it should be understood that othersignals may also be used, for example orthogonal polynomials such asJacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials, Laguerre-Gaussian polynomials, Hermite-Gaussianpolynomials and Ince-Gaussian polynomials. Q-functions are another classof functions that can be employed as a basis for MLO signals.

In quantum mechanics, a coherent state is a state of a quantum harmonicoscillator whose dynamics most closely resemble the oscillating behaviorof a classical harmonic oscillator system. A squeezed coherent state isany state of the quantum mechanical Hilbert space, such that theuncertainty principle is saturated. That is, the product of thecorresponding two operators takes on its minimum value. In embodimentsof an MLO system, operators correspond to time and frequency domainswherein the time-bandwidth product of the signals is minimized. Thesqueezing property of the signals allows scaling in time and frequencydomain simultaneously, without losing mutual orthogonality among thesignals in each layer. This property enables flexible implementations ofMLO systems in various communications systems.

Because signals with different orders are mutually orthogonal, they canbe overlaid to increase the spectral efficiency of a communicationchannel. For example, when n=0, the optimal baseband signal will have atime-bandwidth product of ½, which is the Nyquist Inter-SymbolInterference (ISI) criteria for avoiding ISI. However, signals withtime-bandwidth products of 3/2, 5/2, 7/2, and higher, can be overlaid toincrease spectral efficiency.

An embodiment of an MLO system uses functions based on modified Hermitepolynomials, 4n, and are defined by:

${\psi_{n}\left( {t,\xi} \right)} = {\frac{\left( {\tanh \mspace{14mu} \xi} \right)^{n\text{/}2}}{2^{n\text{/}2}\left( {{n!}\mspace{14mu} \cosh \mspace{14mu} \xi} \right)^{1\text{/}2}}e^{\frac{1}{2}{t^{2}{\lbrack{1 - {\tanh \mspace{14mu} \xi}}\rbrack}}}H_{n}\mspace{14mu} \left( \frac{t}{\sqrt{2\mspace{14mu} \cosh \mspace{14mu} \xi \mspace{14mu} \sinh \mspace{14mu} \xi}} \right)}$

where t is time, and ξ is a bandwidth utilization parameter. Plots ofΨ_(n) for n ranging from 0 to 9, along with their Fourier transforms(amplitude squared), are shown in FIGS. 5A-5K. The orthogonality ofdifferent orders of the functions may be verified by integrating:

∫∫ψ_(n)(t,ξ)ψ_(m)(t,ξ)dtdξ

The Hermite polynomial is defined by the contour integral:

${H_{n}(z)} = {\frac{n!}{2{\pi!}}{\oint{e^{{- t^{2}} + {2t\; 2}}t^{{- n} - 1}{dt}}}}$

where the contour encloses the origin and is traversed in acounterclockwise direction. Hermite polynomials are described inMathematical Methods for Physicists, by George Arfken, for example onpage 416, the disclosure of which is incorporated by reference.

FIGS. 18A-18K illustrate representative MLO signals and their respectivespectral power densities based on the modified Hermite polynomials Ψ_(n)for n ranging from 0 to 9. FIG. 18A shows plots 1801 and 1804. Plot 1801comprises a curve 1827 representing Ψ₀ plotted against a time axis 1802and an amplitude axis 1803. As can be seen in plot 1801, curve 1827approximates a Gaussian curve. Plot 1804 comprises a curve 1837representing the power spectrum of Ψ₀ plotted against a frequency axis1805 and a power axis 1806. As can be seen in plot 1804, curve 1837 alsoapproximates a Gaussian curve. Frequency domain curve 1807 is generatedusing a Fourier transform of time domain curve 1827. The units of timeand frequency on axis 1802 and 1805 are normalized for basebandanalysis, although it should be understood that since the time andfrequency units are related by the Fourier transform, a desired time orfrequency span in one domain dictates the units of the correspondingcurve in the other domain. For example, various embodiments of MLOsystems may communicate using symbol rates in the megahertz (MHz) orgigahertz (GHz) ranges and the non-0 duration of a symbol represented bycurve 1827, i.e., the time period at which curve 1827 is above 0 wouldbe compressed to the appropriate length calculated using the inverse ofthe desired symbol rate. For an available bandwidth in the megahertzrange, the non-0 duration of a time domain signal will be in themicrosecond range.

FIGS. 18B-18J show plots 1807-1824, with time domain curves 1828-1836representing Ψ₁ through Ψ₉, respectively, and their correspondingfrequency domain curves 1838-1846. As can be seen in FIGS. 18A-18J, thenumber of peaks in the time domain plots, whether positive or negative,corresponds to the number of peaks in the corresponding frequency domainplot. For example, in plot 1823 of FIG. 18J, time domain curve 1836 hasfive positive and five negative peaks. In corresponding plot 1824therefore, frequency domain curve 1846 has ten peaks.

FIG. 18K shows overlay plots 1825 and 1826, which overlay curves1827-1836 and 1837-1846, respectively. As indicated in plot 1825, thevarious time domain curves have different durations. However, in someembodiments, the non-zero durations of the time domain curves are ofsimilar lengths. For an MLO system, the number of signals usedrepresents the number of overlays and the improvement in spectralefficiency. It should be understood that, while ten signals aredisclosed in FIGS. 18A-18K, a greater or lesser quantity of signals maybe used, and that further, a different set of signals, rather than theΨ_(n) signals plotted, may be used.

MLO signals used in a modulation layer have minimum time-bandwidthproducts, which enable improvements in spectral efficiency, and arequadratically integrable. This is accomplished by overlaying multipledemultiplexed parallel data streams, transmitting them simultaneouslywithin the same bandwidth. The key to successful separation of theoverlaid data streams at the receiver is that the signals used withineach symbols period are mutually orthogonal. MLO overlays orthogonalsignals within a single symbol period. This orthogonality prevents ISIand inter-carrier interference (ICI).

Because MLO works in the baseband layer of signal processing, and someembodiments use QAM architecture, conventional wireless techniques foroptimizing air interface, or wireless segments, to other layers of theprotocol stack will also work with MLO. Techniques such as channeldiversity, equalization, error correction coding, spread spectrum,interleaving and space-time encoding are applicable to MLO. For example,time diversity using a multipath-mitigating rake receiver can also beused with MLO. MLO provides an alternative for higher order QAM, whenchannel conditions are only suitable for low order QAM, such as infading channels. MLO can also be used with CDMA to extend the number oforthogonal channels by overcoming the Walsh code limitation of CDMA. MLOcan also be applied to each tone in an OFDM signal to increase thespectral efficiency of the OFDM systems.

Embodiments of MLO systems amplitude modulate a symbol envelope tocreate sub-envelopes, rather than sub-carriers. For data encoding, eachsub-envelope is independently modulated according to N-QAM, resulting ineach sub-envelope independently carrying information, unlike OFDM.Rather than spreading information over many sub-carriers, as is done inOFDM, for MLO, each sub-envelope of the carrier carries separateinformation. This information can be recovered due to the orthogonalityof the sub-envelopes defined with respect to the sum of squares overtheir duration and/or spectrum. Pulse train synchronization or temporalcode synchronization, as needed for CDMA, is not an issue, because MLOis transparent beyond the symbol level. MLO addresses modification ofthe symbol, but since CDMA and TDMA are spreading techniques of multiplesymbol sequences over time. MLO can be used along with CDMA and TDMA.

FIG. 19 illustrates a comparison of MLO signal widths in the time andfrequency domains. Time domain envelope representations 1901-1903 ofsignals SH0-SH3 are illustrated as all having a duration T_(S). SH0-SH3may represent PSI₀-PSI₂, or may be other signals. The correspondingfrequency domain envelope representations are 1905-1907, respectively.SH0 has a bandwidth BW, SH1 has a bandwidth three times BW, and SH2 hasa bandwidth of 5BW, which is five times as great as that of SH0. Thebandwidth used by an MLO system will be determined, at least in part, bythe widest bandwidth of any of the signals used. The highest ordersignal must set within the available bandwidth. This will set theparameters for each of the lower order signals in each of the layers andenable the signals to fit together without interference. If each layeruses only a single signal type within identical time windows, thespectrum will not be fully utilized, because the lower order signalswill use less of the available bandwidth than is used by the higherorder signals.

FIG. 20A illustrates a spectral alignment of MLO signals that accountsfor the differing bandwidths of the signals, and makes spectral usagemore uniform, using SH0-SH3. Blocks 2001-2004 are frequency domainblocks of an OFDM signal with multiple subcarriers. Block 2003 isexpanded to show further detail. Block 2003 comprises a first layer 2003x comprised of multiple SH0 envelopes 2003 a-2003 o. A second layer 2003y of SH1 envelopes 2003 p-2003 t has one third the number of envelopesas the first layer. In the illustrated example, first layer 2003 x has15 SH0 envelopes, and second layer 2003 y has five SH1 envelopes. Thisis because, since the SH1 bandwidth envelope is three times as wide asthat of SH0, 15 SH0 envelopes occupy the same spectral width as five SH1envelopes. The third layer 2003 z of block 2003 comprises three SH2envelopes 2003 u-2003 w, because the SH2 envelope is five times thewidth of the SH0 envelope.

The total required bandwidth for such an implementation is a multiple ofthe least common multiple of the bandwidths of the MLO signals. In theillustrated example, the least common multiple of the bandwidth requiredfor SH0, SH1, and SH2 is 15BW, which is a block in the frequency domain.The OFDM-MLO signal can have multiple blocks, and the spectralefficiency of this illustrated implementation is proportional to(15+5+3)/15.

FIGS. 20B-20C illustrate a situation wherein the frequency domainenvelopes 2020-2024 are each located in a separate layer within a samephysical band width 2025. However, each envelope rather than beingcentered on a same center frequency as shown in FIG. 19 has its owncenter frequency 2026-2030 shifted in order to allow a slided overlay.The purposed of the slided center frequency is to allow better use ofthe available bandwidth and insert more envelopes in a same physicalbandwidth.

Since each of the layers within the MLO signal comprises a differentchannel, different service providers may share a same bandwidth by beingassigned to different MLO layers within a same bandwidth. Thus, within asame bandwidth, service provider one may be assigned to a first MLOlayer, service provider two may be assigned to a second MLO layer and soforth.

FIG. 21 illustrates another spectral alignment of MLO signals, which maybe used alternatively to alignment scheme shown in FIG. 20. In theembodiment illustrated in FIG. 21, the OFDM-MLO implementation stacksthe spectrum of SH0, SH1, and SH2 in such a way that the spectrum ineach layer is utilized uniformly. Layer 2100A comprises envelopes2101A-2101D, which includes both SH0 and SH2 envelopes. Similarly, layer2100C, comprising envelopes 2103A-2103D, includes both SH0 and SH2envelopes. Layer 2100B, however, comprising envelopes 2102A-2102D,includes only SH1 envelopes. Using the ratio of envelope sizes describedabove, it can be easily seen that BW+5BW=3BW+3BW. Thus, for each SH0envelope in layer 2100A, there is one SH2 envelope also in layer 2100Cand two SH1 envelopes in layer 2100B.

Three Scenarios Compared:

1) MLO with 3 Layers defined by:

${f_{0} = {W_{0}\mspace{14mu} e^{- \frac{t^{2}}{4}}}},{W_{0} = 0.6316}$${{f_{1}(t)} = {W_{1}\mspace{14mu} t\mspace{14mu} e^{- \frac{t^{2}}{4}}}},{W_{1} \approx 0.6316}$${{f_{2}(t)} = {W_{2}\mspace{14mu} \left( {t^{2} - 1} \right)\mspace{14mu} e^{- \frac{t^{2}}{4}}}},{W_{2} \approx 0.4466}$

(The current FPGA implementation uses the truncation interval of [−6,6].)2) Conventional scheme using rectangular pulse3) Conventional scheme using a square-root raised cosine (SRRC) pulsewith a roll-off factor of 0.5

For MLO pulses and SRRC pulse, the truncation interval is denoted by[−t1, t1] in the following figures. For simplicity, we used the MLOpulses defined above, which can be easily scaled in time to get thedesired time interval (say micro-seconds or nano-seconds). For the SRRCpulse, we fix the truncation interval of [−3T, 3T] where T is the symbolduration for all results presented in this document.

Bandwidth Efficiency

The X-dB bounded power spectral density bandwidth is defined as thesmallest frequency interval outside which the power spectral density(PSD) is X dB below the maximum value of the PSD. The X-dB can beconsidered as the out-of-band attenuation.

The bandwidth efficiency is expressed in Symbols per second per Hertz.The bit per second per Hertz can be obtained by multiplying the symbolsper second per Hertz with the number of bits per symbol (i.e.,multiplying with log 2 M for M-ary QAM).

Truncation of MLO pulses introduces inter-layer interferences (ILI).However, the truncation interval of [−6, 6] yields negligible ILI while[−4, 4] causes slight tolerable ILI. Referring now to FIG. 22, there isillustrated the manner in which a signal, for example a superQAM signal,may be layered to create ILI. FIG. 22 illustrates 3 different superQAMsignals 2202, 2204 and 2206. The superQAM signals 2202-2206 may betruncated and overlapped into multiple layers using QLO in the mannerdescribed herein above. However, as illustrated in FIG. 66, thetruncation of the superQAM signals 2202-2206 that enables the signals tobe layered together within a bandwidth T_(d) 2302 creates a singlesignal 2304 having the interlayer interference between each of thelayers containing a different signal produced by the QLO process. TheILI is caused between a specific bit within a specific layer having aneffect on other bits within another layer of the same symbol.

The bandwidth efficiency of MLO may be enhanced by allowing inter-symbolinterference (ISI). To realize this enhancement, designing transmitterside parameters as well as developing receiver side detection algorithmsand error performance evaluation can be performed. One manner in whichISI may be created is when multilayer signals such as that illustratedin FIG. 23 are overlapped with each other in the manner illustrated inFIG. 24. Multiple signal symbols 2402 are overlapped with each other inorder to enable to enable more symbols to be located within a singlebandwidth. The portions of the signal symbols 2402 that are overlappingcause the creation of ISI. Thus, a specific bit at a specific layer willhave an effect on the bits of nearby symbols.

The QLO transmission and reception system can be designed to have aparticular known overlap between symbols. The system can also bedesigned to calculate the overlaps causing ISI (symbol overlap) and ILI(layer overlay). The ISI and ILI can be expressed in the format of aNM*NM matrix derived from a N*NM matrix. N comprises the number oflayers and M is the number of symbols when considering ISI. Referringnow to FIG. 25, there is illustrated a fixed channel matrix H_(xy) whichis a N*NM matrix. From this we can calculate another matrix which isH_(yx) which is a NM*NM matrix. The ISI and ILI can be canceled by (a)applying a filter of H_(yx) ⁻¹ to the received vector or (b)pre-distorting the transmitted signal by the SVD (singular valuedecomposition) of H_(yx) ⁻¹. Therefore, by determining the matrix H ofthe fixed channel, the signal may be mathematically processed to removeISL and ILI.

When using orthogonal functions such as Hermite Gaussian (HG) functions,the functions are all orthogonal for any permutations of the index ifinfinitely extended. However, when the orthogonal functions aretruncated as discussed herein above, the functions becomepseudo-orthogonal. This is more particularly illustrated in FIG. 26. Inthis case, orthogonal functions are represented along each of the axes.At the intersection of the same orthogonal functions, functions arecompletely correlated and a value of “1” is indicated. Thus, a diagonalof “1” exists with each of the off diagonal elements comprising a “0”since these functions are completely orthogonal with each other. Whentruncated HG choose functions are used the 0 values will not be exactly0 since the functions are no longer orthogonal but arepseudo-orthogonal.

However, the HG functions can be selected in a manner that the functionsare practically orthogonal. This is achieved by selecting the HG signalsin a sequence to achieve better orthogonality. Thus, rather thanselecting the initial three signals in a three signal HG signal sequence(P0 P1 P2), various other sequences that do not necessarily comprise thefirst three signals of the HG sequence may be selected as shown below.

P0 P1 P4 P0 P3 P6 P0 P1 P6 P0 P4 P5 P0 P2 P3 P0 P5 P6 P0 P2 P5 P1 P3 P6P0 P3 P4 P2 P5 P6

Similar selection of sequences may be done to achieve betterorthogonality with two signals, four signals, etc.

The techniques described herein are applicable to a wide variety ofcommunication band environments. They may be applied across the visibleand invisible bands and include RF, Fiber, Freespace optical and anyother communications bands that can benefit from the increased bandwidthprovided by the disclosed techniques.

Application of OAM to Optical Communication

Utilization of OAM for optical communications is based on the fact thatcoaxially propagating light beams with different OAM states can beefficiently separated. This is certainly true for orthogonal modes suchas the LG beam. Interestingly, it is also true for general OAM beamswith cylindrical symmetry by relying only on the azimuthal phase.Considering any two OAM beams with an azimuthal index of

1 and

2, respectively:

U ₁(r,θ,z)=A ₁(r,z)exp(il ₁θ)  (12)

where r and z refers to the radial position and propagation distancerespectively, one can quickly conclude that these two beams areorthogonal in the sense that:

$\begin{matrix}{{\int\limits_{0}^{2\pi}{U_{1}U_{2}^{*}d\; \theta}} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} I_{1}} \neq I_{2}} \\{A_{1}A_{2}^{*}} & {{{if}\mspace{14mu} I_{1}} = I_{2}}\end{matrix} \right.} & (13)\end{matrix}$

There are two different ways to take advantage of the distinctionbetween OAM beams with different l states in communications. In thefirst approach, N different OAM states can be encoded as N differentdata symbols representing “0”, “1”, . . . , “N−1”, respectively. Asequence of OAM states sent by the transmitter therefore represents datainformation. At the receiver, the data can be decoded by checking thereceived OAM state. This approach seems to be more favorable to thequantum communications community, since OAM could provide for theencoding of multiple bits (log 2(N)) per photon due to the infinitelycountable possibilities of the OAM states, and so could potentiallyachieve a higher photon efficiency. The encoding/decoding of OAM statescould also have some potential applications for on-chip interconnectionto increase computing speed or data capacity.

The second approach is to use each OAM beam as a different data carrierin an SDM (Spatial Division Multiplexing) system. For an SDM system, onecould use either a multi-core fiber/free space laser beam array so thatthe data channels in each core/laser beam are spatially separated, oruse a group of orthogonal mode sets to carry different data channels ina multi-mode fiber (MMF) or in free space. Greater than 1 petabit/s datatransmission in a multi-core fiber and up to 6 linearly polarized (LP)modes each with two polarizations in a single core multi-mode fiber hasbeen reported. Similar to the SDM using orthogonal modes, OAM beams withdifferent states can be spatially multiplexed and demultiplexed, therebyproviding independent data carriers in addition to wavelength andpolarization. Ideally, the orthogonality of OAM beams can be maintainedin transmission, which allows all the data channels to be separated andrecovered at the receiver. A typical embodiments of OAM multiplexing isconceptually depicted in FIG. 27. An obvious benefit of OAM multiplexingis the improvement in system spectral efficiency, since the samebandwidth can be reused for additional data channels.

OAM Beam Generation and Detection

Many approaches for creating OAM beams have been proposed anddemonstrated. One could obtain a single or multiple OAM beams directlyfrom the output of a laser cavity, or by converting a fundamentalGaussian beam into an OAM beam outside a cavity. The converter could bea spiral phase plate, diffractive phase holograms, metal materials,cylindrical lens pairs, q-plates or fiber structures. There are alsodifferent ways to detect an OAM beam, such as using a converter thatcreates a conjugate helical phase, or using a plasmonic detector.

Mode Conversion Approaches

Referring now to FIG. 28, among all external-cavity methods, perhaps themost straightforward one is to pass a Gaussian beam through a coaxiallyplaced spiral phase plate (SPP) 2802. An SPP 2802 is an optical elementwith a helical surface, as shown in FIG. 12E. To produce an OAM beamwith a state of

, the thickness profile of the plate should be machined as

λθ/2π(n−1), where n is the refractive index of the medium. A limitationof using an SPP 2802 is that each OAM state requires a differentspecific plate. As an alternative, reconfigurable diffractive opticalelements, e.g., a pixelated spatial light modulator (SLM) 2804, or adigital micro-mirror device can be programmed to function as anyrefractive element of choice at a given wavelength. As mentioned above,a helical phase profile exp(i

θ) converts a linearly polarized Gaussian laser beam into an OAM mode,whose wave front resembles an

-fold corkscrew 2806, as shown at 2804. Importantly, the generated OAMbeam can be easily changed by simply updating the hologram loaded on theSLM 2804. To spatially separate the phase-modulated beam from thezeroth-order non-phase-modulated reflection from the SLM, a linear phaseramp is added to helical phase code (i.e., a “fork”-like phase pattern2808 to produce a spatially distinct first-order diffracted OAM beam,carrying the desired charge. It should also be noted that theaforementioned methods produce OAM beams with only an azimuthal indexcontrol. To generate a pure LG_(l,p) mode, one must jointly control boththe phase and the intensity of the wavefront. This could be achievedusing a phase-only SLM with a more complex phase hologram.

Some novel material structures, such as metal-surface, can also be usedfor OAM generation. A compact metal-surface could be made into a phaseplate by manipulation of the structure caused spatial phase response. Asshown in FIGS. 29A and 29B, a V-shaped antenna array 2902 is fabricatedon the metal surface 2904, each of which is composed of two arms 2906,2908 connected at one end 2910. A light reflected by this plate wouldexperience a phase change ranging from 0 to 2π, determined by the lengthof the arms and angle between two arms. To generate an OAM beam, thesurface is divided into 8 sectors 2912, each of which introduces a phaseshift from 0 to 7π/4 with a step of π/4. The OAM beam with

=+1 is obtained after the reflection, as shown in FIG. 29C.

Referring now to FIG. 30, another interesting liquid crystal-baseddevice named “q-plate” 3002 is also used as a mode converter whichconverts a circularly polarized beam 3004 into an OAM beam 3006. Aq-plate is essentially a liquid crystal slab with a uniform birefringentphase retardation of π and a spatially varying transverse optical axis3008 pattern. Along the path circling once around the center of theplate, the optical axis of the distributed crystal elements may have anumber of rotations defined by the value of q. A circularly polarizedbeam 3004 passing through this plate 3002 would experience a helicalphase change of exp (ilθ) with

=2q, as shown in FIG. 30.

Note that almost all the mode conversion approaches can also be used todetect an OAM beam. For example, an OAM beam can be converted back to aGaussian-like non-OAM beam if the helical phase front is removed, e.g.,by passing the OAM beam through a conjugate SPP or phase hologram.

Intra-Cavity Approaches

Referring now to FIG. 31, OAM beams are essentially higher order modesand can be directly generated from a laser resonator cavity. Theresonator 3100 supporting higher order modes usually produce the mixtureof multiple modes 3104, including the fundamental mode. In order toavoid the resonance of fundamental Gaussian mode, a typical approach isto place an intra-cavity element 3106 (spiral phase plate, tiled mirror)to force the oscillator to resonate on a specific OAM mode. Otherreported demonstrations include the use of an annular shaped beam aslaser pump, the use of thermal lensing, or by using a defect spot on oneof the resonator mirrors.

OAM Beams Multiplexing and Demultiplexing

One of the benefits of OAM is that multiple coaxially propagating OAMbeams with different

states provide additional data carriers as they can be separated basedonly on the twisting wavefront. Hence, one of the critical techniques isthe efficient multiplexing/demultiplexing of OAM beams of different

states, where each carries an independent data channel and all beams canbe transmitted and received using a single aperture pair. Severalmultiplexing and demultiplexing techniques have been demonstrated,including the use of an inverse helical phase hologram to down-convertthe OAM into a Gaussian beam, a mode sorter, free-space interferometers,a photonic integrated circuit, and q-plates. Some of these techniquesare briefly described below.

Beam Splitter and Inverse Phase Hologram

A straightforward way of multiplexing is simply to use cascaded 3-dBbeam splitters (BS) 3202. Each BS 3202 can coaxially multiplex two beams3203 that are properly aligned, and cascaded N BSs can multiplex N+1independent OAM beams at most, as shown in FIG. 32. Similarly, at thereceiver end, the multiplexed beam 3205 is divided into four copies 3204by BS 3202. To demultiplex the data channel on one of the beams (e.g.,with l=l_i), a phase hologram 3206 with a spiral charge of [(−l]_i isapplied to all the multiplexed beams 3204. As a result, the helicalphase on the target beam is removed, and this beam evolves into afundamental Gaussian beam, as shown in FIG. 33. The down-converted beamcan be isolated from the other beams, which still have helical phasefronts by using a spatial mode filter 3308 (e.g., a single mode fiberonly couples the power of the fundamental Gaussian mode due to the modematching theory). Accordingly, each of the multiplexed beams 3304 can bedemultiplexed by changing the spiral phase hologram 3306. Although thismethod is very power-inefficient since the BSs 3302 and the spatial modefilter 3306 cause a lot of power loss, it was used in the initial labdemonstrations of OAM multiplexing/demultiplexing, due to the simplicityof understanding and the reconfigurability provided by programmableSLMs.

Optical Geometrical Transformation-Based Mode Sorter

Referring now to FIG. 34, another method of multiplexing anddemultiplexing, which could be more power-efficient than the previousone (using beam splitters), is the use of an OAM mode sorter. This modesorter usually comprises three optical elements, including a transformer3402, a corrector 3404, and a lens 3406, as shown in FIG. 34. Thetransformer 3402 performs a geometrical transformation of the input beamfrom log-polar coordinates to Cartesian coordinates, such that theposition (x,y) in the input plane is mapped to a new position (u,v) inthe output plane, where

${u = {{- a}\mspace{14mu} {\ln \left( \frac{\sqrt{x^{2} + y^{2}}}{b} \right)}}},$

and v=a arctan(y/x). Here, a and b are scaling constants. The corrector3404 compensates for phase errors and ensures that the transformed beamis collimated. Considering an input OAM beam with a ring-shaped beamprofile, it can be unfolded and mapped into a rectangular-shaped planewave with a tilted phase front. Similarly, multiple OAM beams havingdifferent l states will be transformed into a series of plane waves eachwith a different phase tilt. A lens 3406 focuses these tilted planewaves into spatially separated spots in the focal plane such that allthe OAM beams are simultaneously demultiplexed. As the transformation isreciprocal, if the mode sorter is used in reverse it can become amultiplexer for OAM. A Gaussian beam array placed in the focal plane ofthe lens 3406 is converted into superimposed plane waves with differenttilts. These beams then pass through the corrector and the transformersequentially to produce properly multiplexed OAM beams.

Free Space Communications

The first proof-of-concept experiment using OAM for free spacecommunications transmitted eight different OAM states each representinga data symbol one at a time. The azimuthal index of the transmitted OAMbeam is measured at the receiver using a phase hologram modulated with abinary grating. To effectively use this approach, fast switching isrequired between different OAM states to achieve a high data rate.Alternatively, classic communications using OAM states as data carrierscan be multiplexed at the transmitter, co-propagated through a freespace link, and demultiplexed at a receiver. The total data rate of afree space communication link has reached 100 Tbit/s or even beyond byusing OAM multiplexing. The propagation of OAM beams through a realenvironment (e.g., across a city) is also under investigation.

Basic Link Demonstrations

Referring now to FIGS. 35-36B, initial demonstrates of using OAMmultiplexing for optical communications include free space links using aGaussian beam and an OAM beam encoded with OOK data. Four monochromaticGaussian beams each carrying an independent 50.8 Gbit/s (4×12.7 Gbit/s)16-QAM signal were prepared from an IQ modulator and free-spacecollimators. The beams were converted to OAM beams with

=−8, +10, +12 and −14, respectively, using 4 SLMs each loaded with ahelical phase hologram, as shown in FIG. 30A. After being coaxiallymultiplexed using cascaded 3 dB-beam splitters, the beams werepropagated through ˜1 m distance in free-space under lab conditions. TheOAM beams were detected one at a time, using an inverse helical phasehologram and a fiber collimator together with a SMF. The 16-QAM data oneach channel was successfully recovered, and a spectral efficiency of12.8 bit/s/Hz in this data link was achieved, as shown in FIGS. 36A and36B.

A following experiment doubled the spectral efficiency by adding thepolarization multiplexing into the OAM-multiplexed free-space data link.Four different OAM beams (

=+4, +8, −8, +16) on each of two orthogonal polarizations (eightchannels in total) were used to achieve a Terabit/s transmission link.The eight OAM beams were multiplexed and demultiplexed using the sameapproach as mentioned above. The measured crosstalk among channelscarried by the eight OAM beams is shown in Table 1, with the largestcrosstalk being ˜−18.5 dB. Each of the beams was encoded with a 42.8Gbaud 16-QAM signal, allowing a total capacity of ˜1.4 (42.8×4×4×2)Tbit/s.

TABLE 1 OAM₊₄ OAM₊₈ OAM⁻⁸ OAM₊₁₆ Measured Crosstalk X-Pol. Y-Pol. X-Pol.Y-Pol. X-Pol. Y-Pol. X-Pol. Y-Pol. OAM₊₄ (dB) X-Pol. −23.2 −26.7 −30.8−30.5 −27.7 −24.6 −30.1 Y-Pol. −25.7 OAM₊₈ (dB) X-Pol. −26.6 −23.5 −21.6−18.9 −25.4 −23.9 −28.8 Y-Pol. −25.0 OAM⁻⁸ (dB) X-Pol. −27.5 −33.9 −27.6−30.8 −20.5 −26.5 −21.6 Y-Pol. −26.8 OAM₊₁₆ (dB) X-Pol. −24.5 −31.2−23.7 −23.3 −25.8 −26.1 −30.2 Y-Pol. −24.0 Total from other OAMs *(dB)−21.8 −21.0 −21.2 −21.4 −18.5 −21.2 −22.2 −20.7

The capacity of the free-space data link was further increased to 100Tbit/s by combining OAM multiplexing with PDM (phase divisionmultiplexing) and WDM (wave division multiplexing). In this experiment,24 OAM beams (

=±4, ±7, ±10, ±13, ±16, and ±19, each with two polarizations) wereprepared using 2 SLMs, the procedures for which are shown in FIG. 37A at3702-3706. Specifically, one SLM generated a superposition of OAM beamswith

=+4, +10, and +16, while the other SLM generated another set of threeOAM beams with

=+7, +13, and +19 (FIG. 37A). These two outputs were multiplexedtogether using a beam splitter, thereby multiplexing six OAM beams:

=+4, +7, +10, +13, +16, and +19 (FIG. 37A). Secondly, the sixmultiplexed OAM beams were split into two copies. One copy was reflectedfive times by three mirrors and two beam splitters, to create anothersix OAM beams with inverse charges (FIG. 37B). There was a differentialdelay between the two light paths to de-correlate the data. These twocopies were then combined again to achieve 12 multiplexed OAM beams with

=±4, ±7, ±10, ±13, ±16, and ±19 (FIG. 37B). These 12 OAM beams weresplit again via a beam splitter. One of these was polarization-rotatedby 90 degrees, delayed by ˜33 symbols, and then recombined with theother copy using a polarization beam splitter (PBS), finallymultiplexing 24 OAM beams (with

=±4, ±7, ±10, ±13, ±16, and ±19 on two polarizations). Each of the beamcarried a WDM signal comprising 100 GHz-spaced 42 wavelengths(1,536.34-1,568.5 nm), each of which was modulated with 100 Gbit/s QPSKdata. The observed optical spectrum of the WDM signal carried on one ofthe demultiplexed OAM beams (

=+10).

Atmospheric Turbulence Effects on OAM Beams

One of the critical challenges for a practical free-space opticalcommunication system using OAM multiplexing is atmospheric turbulence.It is known that inhomogeneities in the temperature and pressure of theatmosphere lead to random variations in the refractive index along thetransmission path, and can easily distort the phase front of a lightbeam. This could be particularly important for OAM communications, sincethe separation of multiplexed OAM beams relies on the helicalphase-front. As predicted by simulations in the literature, theserefractive index inhomogeneities may cause inter-modal crosstalk amongdata channels with different OAM states.

The effect of atmospheric turbulence is also experimentally evaluated.For the convenience of estimating the turbulence strength, one approachis to emulate the turbulence in the lab using an SLM or a rotating phaseplate. FIG. 38A illustrates an emulator built using a thin phase screenplate 3802 that is mounted on a rotation stage 3804 and placed in themiddle of the optical path. The pseudo-random phase distributionmachined on the plate 3802 obeys Kolmogorov spectrum statistics, whichare usually characterized by a specific effective Fried coherence lengthr0. The strength of the simulated turbulence 3806 can be varied eitherby changing to a plate 3802 with a different r0, or by adjusting thesize of the beam that is incident on the plate. The resultant turbulenceeffect is mainly evaluated by measuring the power of the distorted beamdistributed to each OAM mode using an OAM mode sorter. It was foundthat, as the turbulence strength increases, the power of the transmittedOAM mode would leak to neighboring modes and tend to be equallydistributed among modes for stronger turbulence. As an example, FIG. 38Bshows the measured average power (normalized) l=3 beam under differentemulated turbulence conditions. It can be seen that the majority of thepower is still in the transmitted OAM mode 3808 under weak turbulence,but it spreads to neighboring modes as the turbulence strengthincreases.

Turbulence Effects Mitigation Techniques

One approach to mitigate the effects of atmospheric turbulence on OAMbeams is to use an adaptive optical (AO) system. The general idea of anAO system is to measure the phase front of the distorted beam first,based on which an error correction pattern can be produced and can beapplied onto the beam transmitter to undo the distortion. As for OAMbeams with helical phase fronts, it is challenging to directly measurethe phase front using typical wavefront sensors due to the phasesingularity. A modified AO system can overcome this problem by sending aGaussian beam as a probe beam to sense the distortion, as shown in FIG.39A. Due to the fact that turbulence is almost independent of the lightpolarization, the probe beam is orthogonally polarized as compared toall other beams for the sake of convenient separation at beam separator3902. The correction phase pattern can be derived based on the probebeam distortion that is directly measured by a wavefront sensor 3804. Itis noted that this phase correction pattern can be used tosimultaneously compensate multiple coaxially propagating OAM beams. FIG.39 at 3910-3980 illustrate the intensity profiles of OAM beams with l=1,5 and 9, respectively, for a random turbulence realization with andwithout mitigation. From the far-field images, one can see that thedistorted OAM beams (upper), up to l=9, were partially corrected, andthe measured power distribution also indicates that the channelcrosstalk can be reduced.

Another approach for combating turbulence effects is to partially movethe complexity of optical setup into the electrical domain, and usedigital signal processing (DSP) to mitigate the channel crosstalk. Atypical DSP method is the multiple-input-multiple-output (MIMO)equalization, which is able to blindly estimate the channel crosstalkand cancel the interference. The implementation of a 4×4 adaptive MIMOequalizer in a four-channel OAM multiplexed free space optical linkusing heterodyne detection may be used. Four OAM beams (l=+2, +4, +6 and+8), each carrying 20 Gbit/s QPSK data, were collinearly multiplexed andpropagated through a weak turbulence emulated by the rotating phaseplate under laboratory condition to introduce distortions. Afterdemultiplexing, four channels were coherently detected and recordedsimultaneously. The standard constant modulus algorithm is employed inaddition to the standard procedures of coherent detection to equalizethe channel interference. Results indicate that MIMO equalization couldbe helpful to mitigate the crosstalk caused by either turbulence orimperfect mode generation/detection, and improve both error vectormagnitude (EVM) and the bit-error-rate (BER) of the signal in anOAM-multiplexed communication link. MIMO DSP may not be universallyuseful as outage could happen in some scenarios involving free spacedata links. For example, the majority power of the transmitted OAM beamsmay be transferred to other OAM states under a strong turbulence withoutbeing detected, in which case MIMO would not help to improve the systemperformance.

OAM Free Space Link Design Considerations

To date, most of the experimental demonstrations of opticalcommunication links using OAM beams took place in the lab conditions.There is a possibility that OAM beams may also be used in a free spaceoptical communication link with longer distances. To design such a datalink using OAM multiplexing, several important issues such as beamdivergence, aperture size and misalignment of two transmitter andreceiver, need to be resolved. To study how those parameters affect theperformance of an OAM multiplexed system, a simulation model wasdescribed by Xie et al, the schematic setup of which is shown in FIG.40. Each of the different collimated Gaussian beams 4002 at the samewavelength is followed by a spiral phase plate 4004 with a unique orderto convert the Gaussian beam into a data-carrying OAM beam. Differentorders of OAM beams are then multiplexed at multiplexor 4006 to form aconcentric-ring-shape and coaxially propagate from transmitter 4008through free space to the receiver aperture located at a certainpropagation distance. Propagation of multiplexed OAM beams isnumerically propagated using the Kirchhoff-Fresnel diffraction integral.To investigate the signal power and crosstalk effect on neighboring OAMchannels, power distribution among different OAM modes is analyzedthrough a modal decomposition approach, which corresponds to the casewhere the received OAM beams are demultiplexed without power loss andthe power of a desired OAM channel is completely collected by itsreceiver 4010.

Beam Divergence

For a communication link, it is generally preferable to collect as muchsignal power as possible at the receiver to ensure a reasonablesignal-to-noise ratio (SNR). Based on the diffraction theory, it isknown that a collimated OAM beam diverges while propagating in freespace. Given the same spot size of three cm at the transmitter, an OAMbeam with a higher azimuthal index diverges even faster, as shown inFIG. 41A. On the other hand, the receiving optical element usually has alimited aperture size and may not be able to collect all of the beampower. The calculated link power loss as a function of receiver aperturesize is shown in FIG. 41B, with different transmission distances andvarious transmitted beam sizes. Unsurprisingly, the power loss of a 1-kmlink is higher than that of a 100-m link under the same transmitted beamsize due to larger beam divergence. It is interesting to note that asystem with a transmitted beam size of 3 cm suffers less power loss thanthat of 1 cm and 10 cm over a 100-m link. The 1-cm transmitted beamdiverges faster than the 3 cm beam due to its larger diffraction.However, when the transmitted beam size is 10 cm, the geometricalcharacteristics of the beam dominate over the diffraction, thus leadinglarger spot size at the receiver than the 3 cm transmitted beam. Atrade-off between the diffraction, geometrical characteristics and thenumber of OAMs of the beam therefore needs to be carefully considered inorder to achieve a proper-size received beam when designing a link.

Misalignment Tolerance

Referring now to FIGS. 42A-42C, besides the power loss due tolimited-size aperture and beam divergence, another issue that needsfurther discussion is the potential misalignment between the transmitterand the receiver. In an ideal OAM multiplexed communication link,transmitter and receiver would be perfectly aligned, (i.e., the centerof the receiver would overlap with the center of the transmitted beam4202, and the receiver plane 4204 would be perpendicular to the lineconnecting their centers, as shown in FIG. 42A). However, due todifficulties in aligning because of substrate distances, and jitter andvibration of the transmitter/receiver platform, the transmitter andreceiver may have relative lateral shift (i.e., lateral displacement)(FIG. 42B) or angular shift (i.e., receiver angular error) (FIG. 42C).Both types of misalignment may lead to degradation of systemperformance.

Focusing on a link distance of 100 m, FIGS. 43A and 43B show the powerdistribution among different OAM modes due to lateral displacement andreceiver angular error when only

=+3 is transmitted with a transmitted beam size of 3 cm. In order toinvestigate the effect of misalignment, the receiver aperture size ischosen to be 10 cm, which could cover the whole OAM beam at thereceiver. As the lateral displacement or receiver angular errorincreases, power leakage to other modes (i.e., channel crosstalk)increases while the power on

=+3 state decreases. This is because larger lateral displacement orreceiver angular causes larger phase profile mismatch between thereceived OAM beams and receiver. The power leakage to

=+1 and

=+5 is greater than that of

=+2 and

=+3 due to their larger mode spacing with respect to

=+3. Therefore, a system with larger mode spacing (which also useshigher order OAM states suffers less crosstalk. However, such a systemmay also have a larger power loss due to the fast divergence of higherorder OAM beams, as discussed above. Clearly, this trade-off betweenchannel crosstalk and power loss shall be considered when choosing themode spacing in a specific OAM multiplexed link.

Referring now to FIG. 44, there is a bandwidth efficiency comparisonversus out of band attenuation (X-dB) where quantum level overlay pulsetruncation interval is [−6,6] and the symbol rate is ⅙. Referring alsoto FIG. 45, there is illustrated the bandwidth efficiency comparisonversus out of band attenuation (X-dB) where quantum level overlay pulsetruncation interval is [−6,6] and the symbol rate is ¼.

The QLO signals are generated from the Physicist's special Hermitefunctions:

${{f_{n}\left( {t,\alpha} \right)} = {\sqrt{\frac{\alpha}{\sqrt{\pi}{n!}2^{n}}}{H_{n}\left( {\alpha \; t} \right)}e^{- \frac{\alpha^{2}t^{2}}{2}}}},{\alpha > 0}$

Note that the initial hardware implementation is using

$\alpha = \frac{1}{\sqrt{2}}$

and for consistency with his part,

$\alpha = \frac{1}{\sqrt{2}}$

is used in all figures related to the spectral efficiency.

Let the low-pass-equivalent power spectral density (PSD) of the combinedQLO signals be X(f) and its bandwidth be B. Here the bandwidth isdefined by one of the following criteria.

ACLR1 (First Adjacent Channel Leakage Ratio) in dBc Equals:

${{ACLR}\; 1} = \frac{\int_{B\text{/}2}^{3B\text{/}2}{{X(f)}\mspace{14mu} {df}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

ACLR2 (Second Adjacent Channel Leakage Ratio) in dBc Equals:

${{ACLR}\; 2} = \frac{\int_{3B\text{/}2}^{5B\text{/}2}{{X(f)}\mspace{14mu} {df}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

Out-of-Band Power to Total Power Ratio is:

$\frac{2{\int_{B\text{/}2}^{\infty}{{X(f)}\mspace{14mu} {df}}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

The Band-Edge PSD in dBc/100 kHz Equals:

$\frac{\int_{B\text{/}2}^{\frac{B}{2} + 10^{5}}{{X(f)}\mspace{14mu} {df}}}{\int_{- \infty}^{\infty}{{X(f)}{df}}}$

Referring now to FIG. 46 there is illustrated a performance comparisonusing ACLR1 and ACLR2 for both a square root raised cosine scheme and amultiple layer overlay scheme. Line 4602 illustrates the performance ofa square root raised cosine 4602 using ACLR1 versus an MLO 4604 usingACLR1. Additionally, a comparison between a square root raised cosine4606 using ACLR2 versus MLO 4608 using ACLR2 is illustrated. Table 2illustrates the performance comparison using ACLR.

TABLE 2 Criteria: ACLR1 ≤ −30 dBc per bandwidth Spectral EfficiencyACLR2 ≤ −43 dBc per bandwidth (Symbol/sec/Hz) Gain SRRC [−8T, 8T] β =0.22 0.8765 1.0 Symbol Duration N 1ayers (Tmol) QLO N = 3 Tmol = 4 1.1331.2926 [−8, 8] N = 4 Tmol = 5 1.094 1.2481 Tmol = 4 1.367 1.5596  N = 10Tmol = 8 1.185 1.3520 Tmol = 7 1.355 1.5459 Tmol = 6 1.580 1.8026 Tmol =5 1.896 2.1631 Tmol = 4 2.371 2.7051

Referring now to FIG. 47, there is illustrated a performance comparisonbetween a square root raised cosine 4702 and a MLO 4704 usingout-of-band power. Referring now also to Table 3, there is illustrated amore detailed comparison of the performance using out-of-band power.

TABLE 3 Performance Comparison Using Out-of-Band Power Criterion:Spectral Efficiency Out-of-band Power/Total Power ≤ −30 dB(Symbol/sec/Hz) Gain SRRC [−8 T, 8 T] β = 0.22 0.861 1.0 Symbol N LayersDuration (Tmol) QLO N = 3 Tmol = 3 1.080 1.2544 [−8, 8] N = 4 Tmol = 51.049 1.2184 Tmol = 4 1.311 1.5226  N = 10 Tmol = 8 1.152 1.3380 Tmol =7 1.311 1.5226 Tmol = 6 1.536 1.7840 Tmol = 5 1.844 2.1417 Tmol = 42.305 2.6771

Referring now to FIG. 48, there is further provided a performancecomparison between a square root raised cosine 4802 and a MLO 4804 usingband-edge PSD. A more detailed illustration of the performancecomparison is provided in Table 4.

TABLE 4 Performance Comparison Using Band-Edge PSD Criterion: SpectralEfficiency Band-Edge PSD = −50 dBc/100 kHz (Symbol/sec/Hz) Gain SRRC [−8T, 8 T] β = 0.22 0.810 1.0 Symbol Duration N Layers (Tmol) QLO N = 3Tmol = 4 0.925 1.1420 [−8, 8] N = 8 Tmol = 5 0.912 1.1259 Tmol = 4 1.141.4074  N = 10 Tmol = 8 1.049 1.2951 Tmol = 7 1.198 1.4790 Tmol = 61.398 1.7259 Tmol = 5 1.678 2.0716 Tmol = 4 2.097 2.5889

Referring now to FIGS. 49 and 50, there are more particularlyillustrated the transmit subsystem (FIG. 49) and the receiver subsystem(FIG. 50). The transceiver is realized using basic building blocksavailable as Commercially Off The Shelf products. Modulation,demodulation and Special Hermite correlation and de-correlation areimplemented on a FPGA board. The FPGA board 5002 at the receiver 5000estimated the frequency error and recovers the data clock (as well asdata), which is used to read data from the analog-to-digital (ADC) board5006. The FGBA board 5000 also segments the digital I and Q channels.

On the transmitter side 4900, the FPGA board 4902 realizes the specialhermite correlated QAM signal as well as the necessary control signalsto control the digital-to-analog (DAC) boards 4904 to produce analog I&Qbaseband channels for the subsequent up conversion within the directconversion quad modulator 4906. The direct conversion quad modulator4906 receives an oscillator signal from oscillator 4908.

The ADC 5006 receives the I&Q signals from the quad demodulator 5008that receives an oscillator signal from 5010.

Neither power amplifier in the transmitter nor an LNA in the receiver isused since the communication will take place over a short distance. Thefrequency band of 2.4-2.5 GHz (ISM band) is selected, but any frequencyband of interest may be utilized.

MIMO uses diversity to achieve some incremental spectral efficiency.Each of the signals from the antennas acts as an independent orthogonalchannel. With QLO, the gain in spectral efficiency comes from within thesymbol and each QLO signal acts as independent channels as they are allorthogonal to one another in any permutation. However, since QLO isimplemented at the bottom of the protocol stack (physical layer), anytechnologies at higher levels of the protocol (i.e. Transport) will workwith QLO. Therefore one can use all the conventional techniques withQLO. This includes RAKE receivers and equalizers to combat fading,cyclical prefix insertion to combat time dispersion and all othertechniques using beam forming and MIMO to increase spectral efficiencyeven further.

When considering spectral efficiency of a practical wirelesscommunication system, due to possibly different practical bandwidthdefinitions (and also not strictly bandlimited nature of actual transmitsignal), the following approach would be more appropriate.

Referring now to FIG. 51, consider the equivalent discrete time system,and obtain the Shannon capacity for that system (will be denoted by Cd).Regarding the discrete time system, for example, for conventional QAMsystems in AWGN, the system will be:

y[n]=a x[n]+w[n]

where a is a scalar representing channel gain and amplitude scaling,x[n] is the input signal (QAM symbol) with unit average energy (scalingis embedded in a), y[n] is the demodulator (matched filter) outputsymbol, and index n is the discrete time index.

The corresponding Shannon capacity is:

C _(d)=log₂(1+|a| ²/σ²)

where σ2 is the noise variance (in complex dimension) and |a|2/σ2 is theSNR of the discrete time system.

Second, compute the bandwidth W based on the adopted bandwidthdefinition (e.g., bandwidth defined by −40 dBc out of band power). Ifthe symbol duration corresponding to a sample in discrete time (or thetime required to transmit C_(d) bits) is T, then the spectral efficiencycan be obtained as:

C/W=C _(d)/(T W)bps/Hz

In discrete time system in AWGN channels, using Turbo or similar codeswill give performance quite close to Shannon limit C_(d). Thisperformance in discrete time domain will be the same regardless of thepulse shape used. For example, using either SRRC (square root raisedcosine) pulse or a rectangle pulse gives the same C_(d) (or C_(d)/T).However, when we consider continuous time practical systems, thebandwidths of SRRC and the rectangle pulse will be different. For atypical practical bandwidth definition, the bandwidth for a SRRC pulsewill be smaller than that for the rectangle pulse and hence SRRC willgive better spectral efficiency. In other words, in discrete time systemin AWGN channels, there is little room for improvement. However, incontinuous time practical systems, there can be significant room forimprovement in spectral efficiency.

Referring now to FIG. 52, there is illustrated a PSD plot (BLANK) ofMLO, modified MLO (MMLO) and square root raised cosine (SRRC). From theillustration in FIG. 30, demonstrates the better localization propertyof MLO. An advantage of MLO is the bandwidth. FIG. 30 also illustratesthe interferences to adjacent channels will be much smaller for MLO.This will provide additional advantages in managing, allocating orpackaging spectral resources of several channels and systems, andfurther improvement in overall spectral efficiency. If the bandwidth isdefined by the −40 dBc out of band power, the within-bandwidth PSDs ofMLO and SRRC are illustrated in FIG. 53. The ratio of the bandwidths isabout 1.536. Thus, there is significant room for improvement in spectralefficiency.

Modified MLO systems are based on block-processing wherein each blockcontains N MLO symbols and each MLO symbol has L layers. MMLO can beconverted into parallel (virtual) orthogonal channels with differentchannel SNRs as illustrated in FIG. 54. The outputs provide equivalentdiscrete time parallel orthogonal channels of MMLO.

Referring now to FIG. 55, there are illustrated four MLO symbols thatare included in a single block 5500. The four symbols 5502-5508 arecombined together into the single block 5500. The adjacent symbols5502-5508 each have an overlapping region 5510. This overlapping region5510 causes intersymbol interference between the symbols which must beaccounted for when processing data streams.

Note that the intersymbol interference caused by pulse overlapping ofMLO has been addressed by the parallel orthogonal channel conversion. Asan example, the power gain of a parallel orthogonal virtual channel ofMMLO with three layers and 40 symbols per block is illustrated in FIG.56. FIG. 56 illustrates the channel power gain of the parallelorthogonal channels of MMLO with three layers and T_(sim)=3. By applyinga water filling solution, an optimal power distribution across theorthogonal channels for a fixed transmit power may be obtained. Thetransmit power on the k^(th) orthogonal channel is denoted by P_(k).Then the discrete time capacity of the MMLO can be given by:

$C_{d} = {\sum\limits_{k = 1}^{k}\; {\log_{2}\mspace{14mu} \left( {1 + \frac{\left. P_{k} \middle| a_{k} \right|^{2}}{\sigma_{k}^{2}}} \right)\mspace{20mu} {bits}\mspace{14mu} {per}\mspace{14mu} {block}}}$

Note that K depends on the number of MLO layers, the number of MLOsymbols per block, and MLO symbol duration.

For MLO pulse duration defined by [−t₁, t₁], and symbol durationT_(mlo), the MMLO block length is:

T _(block)=(N−1)T _(mlo)+2t ₁

Suppose the bandwidth of MMLO signal based on the adopted bandwidthdefinition (ACLR, OBP, or other) is W_(mmlo), then the practicalspectral efficiency of MMLO is given by:

$\frac{C_{d}}{W_{mmlo}\mspace{14mu} T_{block}} = {\frac{1}{W_{mmlo}\mspace{14mu} \left\{ {{\left( {N - 1} \right)\mspace{14mu} T_{mlo}} + {2\mspace{14mu} t_{1}}} \right\}}{\sum\limits_{k = 1}^{K}\; {\log_{2}\mspace{14mu} \left( {1 + \frac{\left. P_{k} \middle| \alpha_{k} \right|^{2}}{\sigma_{k}^{2}}} \right)\mspace{14mu} \frac{bps}{Hz}}}}$

FIGS. 57-58 show the spectral efficiency comparison of MMLO with N=40symbols per block, L=3 layers, T_(mlo)=3, t₁=8, and SRRC with duration[−8T, 8T], T=1, and the roll-off factor β=0.22, at SNR of 5 dB. Twobandwidth definitions based on ACLR1 (first adjacent channel leakagepower ratio) and OBP (out of band power) are used.

FIGS. 59-60 show the spectral efficiency comparison of MMLO with L=4layers. The spectral efficiencies and the gains of MMLO for specificbandwidth definitions are shown in the following tables.

TABLE 5 Spectral Efficiency (bps/Hz) Gain with based on ACLR1 ≤ 30 dBcreference per bandwidth to SRRC SRRC 1.7859 1 MMLO (3 layers, Tmlo = 3)2.7928 1.5638 MMLO (4 layers, Tmlo = 3) 3.0849 1.7274

TABLE 6 Gain with Spectral Efficiency (bps/Hz) reference based on OBP ≤−40 dBc to SRRC SRRC 1.7046 1 MMLO (3 layers, Tmlo = 3) 2.3030 1.3510MMLO (4 layers, Tmlo = 3) 2.6697 1.5662

Referring now to FIGS. 61 and 62, there are provided basic blockdiagrams of low-pass-equivalent MMLO transmitters (FIG. 61) andreceivers (FIG. 62). The low-pass-equivalent MMLO transmitter 6100receives a number of input signals 6102 at a block-based transmitterprocessing 6104. The transmitter processing outputs signals to theSH(L−1) blocks 6106 which produce the I&Q outputs. These signals arethen all combined together at a combining circuit 6108 for transmission.

Within the baseband receiver (FIG. 62) 6200, the received signal isseparated and applied to a series of match filters 6202. The outputs ofthe match filters are then provided to the block-based receiverprocessing block 6204 to generate the various output streams.

Consider a block of N MLO-symbols with each MLO symbol carrying Lsymbols from L layers. Then there are NL symbols in a block. Define c(m,n)=symbol transmitted by the m-th MLO layer at the n-th MLO symbol.Write all NL symbols of a block as a column vector as follows:c=[c(0,0), c(1,0), . . . , c(L−1,0), c(0,1), c(1,1), . . . , c(L−1,1), .. . , c(L−1, N−1)]T. Then the outputs of the receiver matched filtersfor that transmitted block in an AWGN channel, defined by the columnvector y of length NL, can be given as y=H c+n, where H is an NL×NLmatrix representing the equivalent MLO channel, and n is a correlatedGaussian noise vector.

By applying SVD to H, we have H=U D VH where D is a diagonal matrixcontaining singular values. Transmitter side processing using V and thereceiver side processing UH, provides an equivalent system with NLparallel orthogonal channels, (i.e., y=H Vc+n and UH y=Dc+UH n). Theseparallel channel gains are given by diagonal elements of D. The channelSNR of these parallel channels can be computed. Note that by thetransmit and receive block-based processing, we obtain parallelorthogonal channels and hence the ISI issue has be resolved.

Since the channel SNRs of these parallel channels are not the same, wecan apply the optimal Water filling solution to compute the transmitpower on each channel given a fixed total transmit power. Using thistransmit power and corresponding channel SNR, we can compute capacity ofthe equivalent system as given in the previous report.

Issues of Fading, Multipath, and Multi-Cell Interference

Techniques used to counteract channel fading (e.g., diversitytechniques) in conventional systems can also be applied in MMLO. Forslowly-varying multi-path dispersive channels, if the channel impulseresponse can be fed back, it can be incorporated into the equivalentsystem mentioned above, by which the channel induced ISI and theintentionally introduced MMLO ISI can be addressed jointly. For fasttime-varying channels or when channel feedback is impossible, channelequalization needs to be performed at the receiver. A block-basedfrequency-domain equalization can be applied and an oversampling wouldbe required.

If we consider the same adjacent channel power leakage for MMLO and theconventional system, then the adjacent cells' interference power wouldbe approximately the same for both systems. If interference cancellationtechniques are necessary, they can also be developed for MMLO.

Channel fading can be another source of intersymbol interference (ISI)and interlayer interference (ILI). One manner for representingsmall-scale signal fading is the use of statistical models. WhiteGaussian noise may be used to model system noise. The effects ofmultipath fading may be modeled using Rayleigh or Rician probabilitydensity functions. Additive white Gaussian noise (AWGN) may berepresented in the following manner. A received signal is:

r(t)=s(t)+n(t)

where: r(t)=a received signal; s(t)=a transmitted signal; andn(t)=random noise signal

Rayleigh fading functions are useful for predicting bit error rate (BER)any multipath environment. When there is no line of sight (LOS) ordominate received signal, the power the transmitted signal may berepresented by:

${P_{r}(r)} = \left\{ \begin{matrix}{{\frac{r}{\sigma^{2}}e^{\frac{- r^{2}}{2\sigma^{2}}}},} & {r \geq 0} \\{\mspace{76mu} {0,}} & {r < 0}\end{matrix} \right.$

where: σ=rms value of received signal before envelope detection,

-   -   σ=time average power of the received signal before envelope        detection.

In a similar manner, Rician functions may be used in situations wherethere is a line of sight or dominant signal within a transmitted signal.In this case, the power of the transmitted signal can be represented by:

${P_{r}(r)} = \left\{ \begin{matrix}{{\frac{r}{\sigma^{2}}e^{\frac{- {({r^{2} + A^{2}})}}{2\sigma^{2}}}{{II}_{0}\left( \frac{A_{r}}{\sigma^{2}} \right)}},} & {A \geq r \geq 0} \\{\mspace{211mu} {0,}} & {{r < 0}\mspace{45mu}}\end{matrix} \right.$

where A=peak amplitude of LOS component

II₀=modified Bessel Function of the first kind and zero-order

These functions may be implemented in a channel simulation to calculatefading within a particular channel using a channel simulator such asthat illustrated in FIG. 63. The channel simulator 6302 includes aBernoulli binary generator 6304 for generating an input signal that isprovided to a rectangular M-QAM modulator 6306 that generates a QAMsignal at baseband. Multipath fading channel block 6308 uses the Ricianequations to simulate multipath channel fading. The simulated multipathfading channel is provided to a noise channel simulator 6310. The noisechannel simulator 6310 simulates AWGN noise. The multipath fadingchannel simulator 6308 further provides channel state information toarithmetic processing block 6312 which utilizes the simulated multipathfading information and the AWGN information into a signal that isdemodulated at QAM demodulator block 6314. The demodulated simulatedsignal is provided to the doubler block 6316 which is input to a receiveinput of an error rate calculator 6318. The error rate calculator 6318further receives at a transmitter input, the simulated transmissionsignal from the Bernoulli binary generator 6304. The error ratecalculator 6318 uses the transmitter input and the received input toprovide in error rate calculation to a bit error rate block 6320 thatdetermines the channel bit error rate. This type of channel simulationfor determining bit error rate will enable a determination of the amountof QLO that may be applied to a signal in order to increase throughputwithout overly increasing the bit error rate within the channel.

Scope and System Description

This report presents the symbol error probability (or symbol error rate)performance of MLO signals in additive white Gaussian noise channel withvarious inter-symbol interference levels. As a reference, theperformance of the conventional QAM without ISI is also included. Thesame QAM size is considered for all layers of MLO and the conventionalQAM.

The MLO signals are generated from the Physicist's special Hermitefunctions:

${f_{n}\mspace{14mu} \left( {t,\alpha} \right)} = {\sqrt{\frac{\alpha}{\sqrt{\pi}{n\mspace{14mu}!}2^{2}}}H_{n}\mspace{14mu} \left( {\alpha \mspace{14mu} t} \right)e^{- \frac{\alpha^{2}t^{2}}{2}}}$

where Hn(αt) is the n^(th) order Hermite polynomial. Note that thefunctions used in the lab setup correspond to

$\alpha = \frac{1}{\sqrt{2}}$

and, for consistency,

$\alpha = \frac{1}{\sqrt{2}}$

is used in this report.

MLO signals with 3, 4 or 10 layers corresponding to n=0˜2, 0˜3, or 0˜9are used and the pulse duration (the range of t) is [−8, 8] in the abovefunction.

AWGN channel with perfect synchronization is considered.

The receiver consists of matched filters and conventional detectorswithout any interference cancellation, i.e., QAM slicing at the matchedfilter outputs.

${\% \mspace{14mu} {pulse}\text{-}{overlapping}} = {\frac{T_{p} - T_{sym}}{T_{p}} \times 100\mspace{14mu} \%}$

where Tp is the pulse duration (16 in the considered setup) and Tsym isthe reciprocal of the symbol rate in each MLO layer. The consideredcases are listed in the following table.

TABLE 7 % of Pulse Overlapping T_(sym) T_(p)    0% 16 16  12.5% 14 1618.75% 13 16   25% 12 16  37.5% 10 16 43.75% 9 16   50% 8 16 56.25% 7 16 62.5% 6 16   75% 4 16

Derivation of the Signals Used in Modulation

To do that, it would be convenient to express signal amplitude s(t) in acomplex form close to quantum mechanical formalism. Therefore thecomplex signal can be represented as:

ψ(t) = s(t) + j σ(t) where  s(t) ≡ real  signalσ(t) = imaginary  signal  (quadrature)${\sigma (t)} = {{\frac{1}{\tau}{\int\limits_{- \infty}^{\infty}{{s(\tau)}\frac{d\; \tau}{\tau - t}{s(t)}}}} = {{- \frac{1}{\pi}}{\int\limits_{- \infty}^{\infty}{{\sigma (t)}\frac{d\; \tau}{\tau - t}}}}}$

Where s(t) and σ(t) are Hilbert transforms of one another and since σ(t)is quadratures of s(t), they have similar spectral components. That isif they were the amplitudes of sound waves, the ear could notdistinguish one form from the other.

Let us also define the Fourier transform pairs as follows:

${\psi (t)} = {\frac{1}{\pi}{\int\limits_{–\infty}^{\infty}{{\phi (f)}e^{j\; \omega \; t}{df}}}}$${\phi (f)} = {\frac{1}{\pi}{\int\limits_{- \infty}^{\infty}{{\psi (t)}e^{{- j}\; \omega \; t}{dt}}}}$ψ^(*)(t)ψ(t) = [s(t)]² + [σ(t)]² + ⋯ ≡ signal  power

Let's also normalize all moments to M₀

$M_{0} = {\int\limits_{0}^{\tau}{{s(t)}\mspace{14mu} {dt}}}$$M_{0} = {\int\limits_{0}^{\tau}{\phi^{*}\phi \mspace{14mu} {df}}}$

Then the moments are as follows:

$M_{0} = {\int\limits_{0}^{\tau}{{s(t)}\mspace{14mu} {dt}}}$$M_{1} = {\int\limits_{0}^{\tau}{t\mspace{14mu} {s(t)}\mspace{14mu} {dt}}}$$M_{2} = {\int\limits_{0}^{\tau}{t^{2}\mspace{14mu} {s(t)}\mspace{14mu} {dt}}}$$M_{N - 1} = {\int\limits_{0}^{\tau}{t^{N - 1}\mspace{14mu} {s(t)}\mspace{14mu} {dt}}}$

In general, one can consider the signal s(t) be represented by apolynomial of order N, to fit closely to s(t) and use the coefficient ofthe polynomial as representation of data. This is equivalent tospecifying the polynomial in such a way that its first N “moments” M_(j)shall represent the data. That is, instead of the coefficient of thepolynomial, we can use the moments. Another method is to expand thesignal s(t) in terms of a set of N orthogonal functions φ_(k)(t),instead of powers of time. Here, we can consider the data to be thecoefficients of the orthogonal expansion. One class of such orthogonalfunctions are sine and cosine functions (like in Fourier series).

Therefore we can now represent the above moments using the orthogonalfunction w with the following moments:

$\overset{\_}{t} = \frac{\int{{\psi^{*}(t)}t\; {\psi (t)}\mspace{14mu} {dt}}}{\int{{\psi^{*}(t)}\mspace{14mu} {\psi (t)}\mspace{14mu} {dt}}}$$\overset{\_}{t^{2}} = \frac{\int{{\psi^{*}(t)}t^{2}\mspace{14mu} {\psi (t)}\mspace{14mu} {dt}}}{\int{{\psi^{*}(t)}\mspace{14mu} {\psi (t)}\mspace{14mu} {dt}}}$$\overset{\_}{t^{n}} = \frac{\int{{\psi^{*}(t)}t^{n}\mspace{14mu} {\psi (t)}\mspace{14mu} {dt}}}{\int{{\psi^{*}(t)}\mspace{14mu} {\psi (t)}\mspace{14mu} {dt}}}$

Similarly,

$\overset{\_}{f} = \frac{\int{{\phi^{*}(f)}f\; {\phi (f)}\mspace{14mu} {df}}}{\int{{\phi^{*}(f)}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}}$$\overset{\_}{f^{2}} = \frac{\int{{\phi^{*}(f)}f^{2}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}}{\int{{\phi^{*}(f)}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}}$$\overset{\_}{f^{n}} = \frac{\int{{\phi^{*}(f)}f^{n}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}}{\int{{\phi^{*}(f)}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}}$

If we did not use complex signal, then:

f=0

To represent the mean values from time to frequency domains, replace:

ϕ(f) → ψ(t)$\left. f\rightarrow{\frac{1}{2\pi \; j}\frac{d}{dt}} \right.$

These are equivalent to somewhat mysterious rule in quantum mechanicswhere classical momentum becomes an operator:

$\left. P_{x}\rightarrow{\frac{h}{2\pi \; j}\frac{\partial}{\partial x}} \right.$

Therefore using the above substitutions, we have:

$\overset{\_}{f} = {\frac{\int{{\phi^{*}(f)}\mspace{14mu} f\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}}{\int{{\phi^{*}(f)}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}} = {\frac{\int{{\psi^{*}(t)}\left( \frac{1}{2\pi \; j} \right)\frac{d\; {\psi (t)}}{dt}{dt}}}{\int{{\psi^{*}(t)}\mspace{14mu} {\psi (t)}\mspace{14mu} {dt}}} = {\left( \frac{1}{2\pi \; j} \right)\frac{\int{\psi^{*}\mspace{14mu} \frac{d\; \psi}{dt}{dt}}}{\int{\psi^{*}\psi \; {dt}}}}}}$

And:

$\overset{\_}{f^{2}} = {\frac{\int{{\phi^{*}(f)}\mspace{14mu} f^{2}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}}{\int{{\phi^{*}(f)}\mspace{14mu} {\phi (f)}\mspace{14mu} {df}}} = {\frac{\int{{\psi^{*}\left( \frac{1}{2\pi \; j} \right)}^{2}\frac{d^{2}}{{dt}^{2}}\psi \mspace{14mu} {dt}}}{\int{\psi^{*}\psi \mspace{14mu} {dt}}} = {{- \left( \frac{1}{{2\pi}\;} \right)^{2}}\frac{\int{\psi^{*}\mspace{14mu} \frac{d^{2}}{{dt}^{2}}\psi \mspace{14mu} {dt}}}{\int{\psi^{*}\psi \mspace{14mu} {dt}}}}}}$$\mspace{76mu} {\overset{\_}{t^{2}} = \frac{\int{\psi^{*}\mspace{14mu} t^{2}\mspace{14mu} \psi \mspace{14mu} {dt}}}{\int{\psi^{*}\psi \mspace{14mu} {dt}}}}$

We can now define an effective duration and effective bandwidth as:

Δt=√{square root over (2π(t−t)² )}=2π·rms in time

Δt=√{square root over (2π(f−f)² )}=2π·rms in frequency

But we know that:

(t−t)² = t ² −( t )²

(f−f)² = f ² −( f )²

We can simplify if we make the following substitutions:

τ=t−t

Ψ(τ)=ψ(t)e ^(−jωτ)

ω₀=ω=2π f=2πf ₀

We also know that:

(Δt)²(Δf)²=(ΔtΔf)²

And therefore:

$\left( {\Delta \; t\mspace{14mu} \Delta \; f} \right)^{2} = {{\frac{1}{4}\left\lbrack {4\frac{\int{\Psi^{*}\mspace{14mu} (\tau)\tau^{2}\mspace{14mu} {\Psi (\tau)}\mspace{14mu} d\; \tau \mspace{14mu} {\int{\frac{d\; \Psi^{*}}{d\; \tau}\frac{d\; \Psi}{d\; \tau}d\; \tau}}}}{\left( {\int{{\Psi^{*}(\tau)}\mspace{14mu} {\psi (\tau)}\mspace{14mu} d\; \tau}} \right)^{2}}} \right\rbrack} \geq \left( \frac{1}{4} \right)}$$\left( {\Delta \; t\mspace{14mu} \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

Now instead of

$\left( {\Delta \; t\; \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

we are interested to force the equality

$\left( {\Delta \; t\; \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

and see what signals satisfy the equality. Given the fixed bandwidth Δf,the most efficient transmission is one that minimizes the time-bandwidthproduct

$\left( {\Delta \; t\; \Delta \; f} \right) \geq \left( \frac{1}{2} \right)$

For a given bandwidth Δf, the signal that minimizes the transmission inminimum time will be a Gaussian envelope. However, we are often givennot the effective bandwidth, but always the total bandwidth f₂−f₁. Now,what is the signal shape which can be transmitted through this channelin the shortest effective time and what is the effective duration?

${\Delta \; t} = {\frac{\frac{1}{\left( {2\pi} \right)^{2}}{\int_{f_{1}}^{f_{2}}{\frac{d\; \phi^{*}}{df}\frac{d\; \phi}{df}}}}{\int_{f_{1}}^{f_{2}}{\phi^{*}\phi \; {df}}}->\min}$

Where φ(f) is zero outside the range f₂−f₁.

To do the minimization, we would use the calculus of variations(Lagrange's Multiplier technique). Note that the denominator is constantand therefore we only need to minimize the numerator as:

$\mspace{20mu} {{{\Delta \; t}->{\min->{\delta \; {\int_{f_{1}}^{f_{2}}{\left( {{\frac{d\; \phi^{*}}{df}\frac{d\; \phi}{df}} + {\Lambda \; \phi^{*}\phi}} \right){df}}}}}} = 0}$  First  Trem${\delta \; {\int_{f_{1}}^{f_{2}}{\frac{d\; \phi^{*}}{df}\frac{d\; \phi}{df}{df}}}} = {{\int{\left( {{\frac{d\; \phi^{*}}{df}\delta \; \frac{d\; \phi}{df}} + {\frac{d\; \phi}{df}\delta \; \frac{d\; \phi^{*}}{df}}} \right){df}}} = {{\int{\left( {{\frac{d\; \phi^{*}}{df}\frac{d\; \delta \; \phi}{df}} + {\frac{d\; \phi}{df}\frac{d\; \delta \; \phi^{*}}{df}}} \right){df}}} = {{\left\lbrack {{\frac{d\; \phi^{*}}{df}\delta \; \phi} + {\frac{d\; \phi}{df}\delta \; \phi^{*}}} \right\rbrack_{f_{1}}^{f_{2}} - {\int{\left( {{\frac{d^{2}\phi^{*}}{{df}^{2}}\delta \; \phi} + {\frac{d^{2}\phi}{{df}^{2}}\delta \; \phi^{*}}} \right){df}}}} = {\int{\left( {{\frac{d^{2}\phi^{*}}{{df}^{2}}\delta \; \phi} + {\frac{d^{2}\phi}{{df}^{2}}\delta \; \phi^{*}}} \right){df}}}}}}$  Second  Trem  δ ∫_(f₁)^(f₂)(Λ ϕ^(*)ϕ)df = Λ ∫_(f₁)^(f₂)(ϕ^(*)δ ϕ + ϕ δ ϕ^(*))df$\mspace{20mu} {{{Both}\mspace{14mu} {Trems}}\mspace{20mu} = {{\int{\left\lbrack {{\left( {\frac{d^{2}\; \phi^{*}}{{df}^{2}} + {\Lambda \; \phi^{*}}} \right)\delta \; \phi} + {\left( {\frac{d^{2}\phi}{{df}^{2}} + {\Lambda \; \phi}} \right)\delta \mspace{2mu} \phi^{*}}} \right\rbrack {df}}} = 0}}$

This is only possible if and only if.

$\left( {\frac{d^{2}\phi}{{df}^{2}} + {\Lambda \; \phi}} \right) = 0$

The solution to this is of the form

${\phi (f)} = {\sin \; k\; {\pi \left( \frac{f - f_{1}}{f_{2} - f_{1}} \right)}}$

Now if we require that the wave vanishes at infinity, but still satisfythe minimum time-bandwidth product:

$\left( {\Delta \; t\; \Delta \; f} \right) = \left( \frac{1}{2} \right)$

Then we have the wave equation of a Harmonic Oscillator:

${\frac{d^{2}\; {\Psi (\tau)}}{d\; \tau^{2}} + {\left( {\lambda - {\alpha^{2}\tau^{2}}} \right){\Psi (\tau)}}} = 0$

which vanishes at infinity only if:

λ = α(2n + 1)$\psi_{n} = {{e^{{- \frac{1}{2}}\omega^{2}\tau^{2}}\frac{d^{n}}{d\; \tau^{n}}e^{{- \alpha^{2}}\tau^{2}}} \propto {H_{n}(\tau)}}$

Where H_(n)(τ) is the Hermit functions and:

$\left( {\Delta \; t\; \Delta \; f} \right) = {\frac{1}{2}\left( {{2n} + 1} \right)}$

So Hermit functions H_(n)(τ) occupy information blocks of ½, 3/2, 5/2, .. . with ½ as the minimum information quanta.

Squeezed States

Here we would derive the complete Eigen functions in the mostgeneralized form using quantum mechanical approach of Dirac algebra. Westart by defining the following operators:

$b = {\sqrt{\frac{m\; \omega^{\prime}}{2\eta}}\left( {x + \frac{ip}{m\; \omega^{\prime}}} \right)}$$b^{+} = {{\sqrt{\frac{m\; \omega^{\prime}}{2\; \eta}}{\left( {x - \frac{ip}{m\; \omega^{\prime}}} \right)\left\lbrack {b,b^{+}} \right\rbrack}} = 1}$a = λ b − μ b⁺ a⁺ = λ b⁺ − μ b

Now we are ready to define Δx and Δp as:

$\left( {\Delta \; x} \right)^{2} = {{\frac{\eta}{2m\; \omega}\left( \frac{\omega}{\omega^{\prime}} \right)} = {\frac{\eta}{2m\; \omega}\left( {\lambda - \mu} \right)^{2}}}$$\left( {\Delta \; p} \right)^{2} = {{\frac{\eta \; m\; \omega}{2}\left( \frac{\omega^{\prime}}{\omega} \right)} = {\frac{\eta \; m\; \omega}{2}\left( {\lambda + \mu} \right)^{2}}}$${\left( {\Delta \; x} \right)^{2}\left( {\Delta \; p} \right)^{2}} = {\frac{\eta^{2}}{4}\left( {\lambda^{2} - \mu^{2}} \right)^{2}}$${\Delta \; x\; \Delta \; p} = {{\frac{\eta}{2}\left( {\lambda^{2} - \mu^{2}} \right)} = \frac{\eta}{2}}$

Now let parameterize differently and instead of two variables λ and μ,we would use only one variable ξ as follows:

λ=sin h ξ

μ=cos h ξ

λ+μ=e ^(ξ)

λ−μ=−e ^(−ξ)

Now the Eigen states of the squeezed case are:

bβ⟩ = ββ⟩ (λ a + μ a⁺)β⟩ = ββ⟩ b = UaU⁺U = e^(ξ/2(a² − a^(+²))) U⁺(ξ)aU(ξ) = a cosh  ξ − a⁺sinh  ξU⁺(ξ)a⁺U(ξ) = a⁺cosh  ξ − a sinh  ξ

We can now consider the squeezed operator:

α, ξ⟩ = U(ξ)D(α)0⟩${D(\alpha)} = {e^{\frac{- {\alpha }^{2}}{2}}e^{\alpha \; a^{+}}e^{{- \alpha^{*}}a}}$${\alpha\rangle} = {\sum\limits_{n = 0}^{\infty}{\frac{\alpha^{n}}{\sqrt{{n!}\;}}e^{\frac{- {\alpha }^{2}}{2}}{n\rangle}}}$${\alpha\rangle} = {e^{\frac{- {\alpha }^{2}}{2} + {\alpha \; a^{+}}}{0\rangle}}$

For a distribution P(n) we would have:

P(n) = ⟨nβ, ξ⟩²${\langle{{\alpha {}\beta},\xi}\rangle} = {\sum\limits_{n = 0}^{\infty}{\frac{\alpha^{n}}{\sqrt{n!}}e^{\frac{- {\alpha }^{2}}{2}}{\langle{{n{}\beta},\xi}\rangle}}}$$e^{{2{zt}} - t^{2}} = {\sum\limits_{n = 0}^{\infty}\frac{{H_{n}(z)}t^{n}}{n!}}$

Therefore the final result is:

${\langle{{n{}\beta},\xi}\rangle} = {\frac{\left( {\tanh \; \xi} \right)^{n/2}}{2^{n/2}\left( {{n!}\cosh \; \xi} \right)^{2}}e^{{{- 1}/2}{({{\beta }^{2} - {\beta^{2}\tanh \; \xi}})}}{H_{n}\left( \frac{\beta}{2\; \sinh \; \xi \; \cosh \; \xi} \right)}}$

Another issue of concern with the use of QLO with QAM is a desire toimprove bit error rate (BER) performance without impacting theinformation rate or bandwidth requirements of the queue a low signal.One manner for improving BER performance utilizes two separateoscillators that are separated by a known frequency Δf. Signalsgenerated in this fashion will enable a determination of the BER.Referring now to FIG. 64, there is illustrated the generation of two bitstreams b1 and B2 that are provided to a pair of QAM modulators 6402 and6404 by a transmitter 6400. Modulator 6402 receives a first carrierfrequency F1 and modulator 6404 receives a second carrier frequency F2.The frequencies F1 and at two are separated by a known value Δf. Thesignals for each modulator are generated and combined at a summingcircuit 6406 to provide the output s(t). The variables in the outputs ofthe QAM modulators are A_(i) (amplitude), f_(i) (frequency) and ϕ_(i)(phase).

Therefore, each constituent QAM modulation occupies a bandwidth:

${BW} = {r_{s} = {\frac{r_{b}}{\log_{2}m}{symbols}\text{/}\sec}}$

where r_(s) equals the symbol rate of each constituent QAM signal.

The total bandwidth of signal s(t) is:

$W = {{r_{s}\left( {1 + \frac{\Delta \; f}{r_{s}}} \right)} = {r_{s} + {\Delta \; f\mspace{11mu} H_{z}}}}$

Therefore, the spectral efficiency η of this two oscillator system is:

$\eta = \frac{2r_{b}}{W}$

but

r_(b) = r₂log₂m$\eta = {\frac{2r_{b}}{W} = {\frac{2r_{s}\log_{2}m}{r_{s}\left( {1 + \frac{\Delta \; f}{r_{s}}} \right)} = {\frac{2\; \log_{2}m}{1 + \frac{\Delta \; f}{r_{s}}}\mspace{14mu} {bits}\text{/}\sec \text{/}{Hz}}}}$

The narrowband noise over the signal s(t) is:

n(t)=n _(I)(t)cos(2πf ₀ t)−n _(q)(t)sin(2πf ₀ t)

Where: n_(I)(t)=noise in I

N _(q)(t)=noise inQ

Each noise occupies a bandwidth of W [Hz] and the average power of eachcomponent is N₀W. N₀ is the noise power spectral density in Watts/Hz.The value of f₀ is the mean value of f₁ and f₂.

Referring now to FIG. 65, there is illustrated a receiver side blockdiagram for demodulating the signal generated with respect to FIG. 65.The received signal s(t)+n(t) is provided to a number of cosine filters6502-6508. Cosine filters 6502 and 6504 filter with respect to carrierfrequency f₁ and cosine filters 6506 and 6508 filter the received signalfor carrier frequency f₂. Each of the filters 6502-6508 provide anoutput to a switch 6510 that provides a number of output to atransformation block 6512. Transformation block 6512 provides two outputsignals having a real portion and an imaginary portion. Each of the realand imaginary portions associated with a signal are provided to anassociated decoding circuit 6514, 6516 to provide the decoded signals b₁and b₂.

$\begin{bmatrix}{a\left( T_{s} \right)} \\{b\left( T_{s} \right)} \\{c\left( T_{s} \right)} \\{d\left( T_{s} \right)}\end{bmatrix} = {{{T_{s}\begin{bmatrix}1 & 0 & K_{1} & K_{2} \\0 & 1 & {- K_{2}} & K_{1} \\K_{1} & {- K_{2}} & 1 & 0 \\K_{2} & K_{1} & 0 & 1\end{bmatrix}}\begin{bmatrix}{A_{1}\left( {\cos \; \phi_{1}} \right)} \\{A_{1}\left( {\sin \; \phi_{1}} \right)} \\{A_{2}\left( {\cos \; \phi_{2}} \right)} \\{A_{2}\left( {\sin \; \phi_{2}} \right)}\end{bmatrix}} + \begin{bmatrix}{N_{I\; 1}\left( T_{s} \right)} \\{N_{Q\; 1}\left( T_{s} \right)} \\{N_{I\; 2}\left( T_{s} \right)} \\{N_{Q\; 2}\left( T_{s} \right)}\end{bmatrix}}$ A⟩    S⟩  N⟩

-   -   (nonsingular so it has        ⁻¹)

|A>=T _(S)

|S>+|N>

Where

${N_{I_{2 +}^{1 -}}\left( T_{s} \right)} = {{\int\limits_{0}^{T_{s}}{{\eta_{s}(t)}{\cos \left( {\frac{2\eta \; \Delta \; f}{2}t} \right)}}} \mp {{\eta_{G}(t)}{\sin \left( {\frac{2\eta \; \Delta \; f}{2}t} \right)}{dt}}}$${N_{Q_{2 +}^{1 -}}\left( T_{s} \right)} = {{\int\limits_{0}^{T_{s}}{{\eta_{I}(t)}{{din}\left( {\frac{2\eta \; \Delta \; f}{2}t} \right)}}} \mp {{\eta_{G}(t)}{\cos \left( {\frac{2\eta \; \Delta \; f}{2}t} \right)}{dt}}}$A⟩  = T_(s)    S⟩ + N⟩

Multiply by

$\frac{1}{T_{s}}^{- 1}$

${\frac{1}{T_{s}}^{- 1}{A\rangle}}\; = \; {{{S\rangle} + {\frac{1}{T_{s}}^{- 1}{N\rangle}}} = {{S\rangle} + {{{\overset{\sim}{N}{Output}\mspace{14mu} {\rangle}\mspace{20mu} {{\overset{\sim}{N}\rangle}\begin{bmatrix}{I_{1}\left( T_{s} \right)} \\{Q_{1}\left( T_{s} \right)} \\{I_{2}\left( T_{s} \right)} \\{Q_{2}\left( T_{s} \right)}\end{bmatrix}}} = {\begin{bmatrix}{A_{1}\left( {\cos \; \phi_{1}} \right)} \\{A_{1}\left( {\sin \; \phi_{1}} \right)} \\{A_{2}\left( {\cos \; \phi_{2}} \right)} \\{A_{2}\left( {\sin \; \phi_{2}} \right)}\end{bmatrix} + {\begin{bmatrix}{{\overset{\sim}{N}}_{I\; 1}\left( T_{s} \right)} \\{{\overset{\sim}{N}}_{Q\; 1}\left( T_{s} \right)} \\{{\overset{\sim}{N}}_{I\; 2}\left( T_{s} \right)} \\{{\overset{\sim}{N}}_{Q\; 2}\left( T_{s} \right)}\end{bmatrix}{\rangle}\mspace{14mu} {S\rangle}\mspace{14mu} {\overset{\sim}{N}\rangle}}}}}}}$

Then the probability of correct decision P_(e) is

P _(e)≈(1−P _(e))⁴≈1−4P _(e) for P _(e)<<1

P_(e)=well known error probability in one dimension for each constituentm-QAM modulation.Therefore, one can calculate BER.

P_(e) comprises the known error probability in one dimension for eachconstituent member of the QAM modulation. Using the known probabilityerror the bit error rate for the channel based upon the known differencebetween frequencies f₁ and f₂ may be calculated.

Adaptive Processing

The processing of signals using QLO may also be adaptively selected tocombat channel impairments and interference. The process for adaptiveQLO is generally illustrated in FIG. 66. First at step 6602 an analysisof the channel environment is made to determine the present operatingenvironment. The level of QLO processing is selected at step 6604 basedon the analysis and used to configure communications. Next, at step6606, the signals are transmitted at the selected level of QLOprocessing. Inquiry step 6608 determines if sufficient channel qualityhas been achieved. If so, the system continues to transmit and theselected QLO processing level at step 6606. If not, control passes backto step 6602 to adjust the level of QLO processing to achieve betterchannel performance.

The processing of signals using mode division multiplexing (MDM) mayalso be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveMDM is generally illustrated in FIG. 67. First at step 6702 an analysisof the channel environment is made to determine the present operatingenvironment. The level of MDM processing is selected at step 6704 basedon the analysis and used to configure communications. Next, at step6706, the signals are transmitted at the selected level of MDMprocessing. Inquiry step 6708 determines if sufficient channel qualityhas been achieved. If so, the system continues to transmit and theselected MDM processing level at step 6706. If not, control passes backto step 6702 to adjust the level of MDM processing to achieve betterchannel performance.

The processing of signals using an optimal combination of QLO and MDMmay also be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveQLO and MDM is generally illustrated in FIG. 68. First at step 6802 ananalysis of the channel environment is made to determine the presentoperating environment. A selected combination of a level of QLO processand a level of MDM processing are selected at step 6804 based on theanalysis and used to configure communications. Next, at step 6806, thesignals are transmitted at the selected level of QLO and MDM processing.Inquiry step 6808 determines if sufficient channel quality has beenachieved. If so, the system continues to transmit and the selectedcombination of QLO and MDM processing levels at step 6806. If not,control passes back to step 6802 to adjust the levels of QLO and MDMprocessing to achieve better channel performance. Adjustments throughthe steps continue until a most optimal combination of QLO and MDMprocessing is achieved to maximize spectral efficiency using a2-dimensional optimization.

The processing of signals using an optimal combination of QLO and QAMmay also be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveQLO and QAM is generally illustrated in FIG. 69. First at step 6902 ananalysis of the channel environment is made to determine the presentoperating environment. A selected combination of a level of QLO processand a level of QAM processing are selected at step 6904 based on theanalysis and used to configure communications. Next, at step 6906, thesignals are transmitted at the selected level of QLO and QAM processing.Inquiry step 6908 determines if sufficient channel quality has beenachieved. If so, the system continues to transmit and the selectedcombination of QLO and QAM processing levels at step 6906. If not,control passes back to step 6902 to adjust the levels of QLO and QAMprocessing to achieve better channel performance. Adjustments throughthe steps continue until a most optimal combination of QLO and QAMprocessing is achieved to maximize spectral efficiency using a2-dimensional optimization.

The processing of signals using an optimal combination of QLO, MDM andQAM may also be adaptively selected to combat channel impairments andinterference and maximize spectral efficiency. The process for adaptiveQLO, MDM and QAM is generally illustrated in FIG. 70. First at step 7002an analysis of the channel environment is made to determine the presentoperating environment. A selected combination of a level of QLOprocessing, a level of MDM processing and a level of QAM processing areselected at step 7004 based on the analysis and used to configurecommunications. Next, at step 7006, the signals are transmitted at theselected level of QLO, MDM and QAM processing. Inquiry step 7008determines if sufficient channel quality has been achieved. If so, thesystem continues to transmit and the selected combination of QLO, MDMand QAM processing levels at step 7006. If not, control passes back tostep 7002 to adjust the levels of QLO, MDM and QAM processing to achievebetter channel performance. Adjustments through the steps continue untila most optimal combination of QLO, MDM and QAM processing is achieved tomaximize spectral efficiency using a 3-dimensional optimization.

The adaptive approaches described herein above may be used with anycombination of QLO, MDM and QAM processing in order to achieve optimalchannel efficiency. In another application distinct modal combinationsmay also be utilized.

Improvement of Pilot Signal Modulation

The above described QLO, MDM and QAM processing techniques may also beused to improve the manner in which a system deals with noise, fadingand other channel impairments by the use of pilot signal modulationtechniques. As illustrated in FIG. 71, a pilot signal 7102 istransmitted between a transmitter 7104 to a receiver 7106. The pilotsignal includes an impulse signal that is received, detected andprocessed at the receiver 7106. Using the information received from thepilot impulse signal, the channel 7108 between the transmitter 7104 andreceiver 7106 may be processed to remove noise, fading and other channelimpairment issues from the channel 7108.

This process is generally described with respect to the flowchart ofFIG. 72. The pilot impulse signal is transmitted at 7202 over thetransmission channel. The impulse response is detected at step 7204 andprocessed to determine the impulse response over the transmissionchannel. Effects of channel impairments such as noise and fading may becountered by multiplying signals transmitted over the transmissionchannel by the inverse of the impulse response at step 7206 in order tocorrect for the various channel impairments that may be up on thetransmission channel. In this way the channel impairments arecounteracted and improved signal quality and reception may be providedover the transmission channel.

Power Control

Adaptive power control may be provided on systems utilizing QLO, MDM andQAM processing to also improve channel transmission. Amplifiernonlinearities within the transmission circuitry and the receivercircuitry will cause impairments in the channel response as moreparticularly illustrated in FIG. 73. As can be seen the channelimpairments and frequency response increase and decrease over frequencyas illustrated generally at 7302. By adaptively controlling the power ofa transmitting unit or a receiving unit and inverse frequency responsesuch as that generated at 7304 may be generated. Thus, when the normalfrequency response 7302 and the inverse frequency response 7304 arecombined, a consistent response 7306 is provided by use of the adaptivepower control.

Backward and Forward Channel Estimation

QLO techniques may also be used with forward and backward channelestimation processes when communications between a transmitter 7402 anda receiver 7404 do not have the same channel response over both theforward and backward channels. As shown in FIG. 74, the forward channel7406 and backward channel 7408 between a transmitter 7402 and receiver7404 may each be processed to determine their channel impulse responses.Separate forward channel estimation response and backward channelestimation response may be used for processing QLO signals transmittedover the forward channel 7406 and backward channel 7408. The differencesin the channel response between the forward channel 7406 and thebackward channel 7408 may arise from differences in the topography ornumber of buildings located within the area of the transmitter 7402 andthe receiver 7404. By treating each of the forward channel 7406 and abackward channel 7408 differently better overall communications may beachieved.

Using MIMO Techniques with QLO

MIMO techniques may be used to improve the performance of QLO-basedtransmission systems. MIMO (multiple input and multiple output) is amethod for multiplying the capacity of a radio link using multipletransmit and receive antennas to exploit multipath propagation. MIMOuses multiple antennas to transmit a signal instead of only a singleantenna. The multiple antennas may transmit the same signal usingmodulation with the signals from each antenna modulated by differentorthogonal signals such as that described with respect to the QLOmodulation in order to provide an improved MIMO based system.

Diversions within OAM beams may also be reduced using phased arrays. Byusing multiple transmitting elements in a geometrical configuration andcontrolling the current and phase for each transmitting element, theelectrical size of the antenna increases as does the performance of theantenna. The antenna system created by two or more individual intendedelements is called an antenna array. Each transmitting element does nothave to be identical but for simplification reasons the elements areoften alike. To determine the properties of the electric field from anarray the array factor (AF) is utilized.

The total field from an array can be calculated by a superposition ofthe fields from each element. However, with many elements this procedureis very unpractical and time consuming. By using different kinds ofsymmetries and identical elements within an array, a much simplerexpression for the total field may be determined. This is achieved bycalculating the so-called array factor (AF) which depends on thedisplacement (and shape of the array), phase, current amplitude andnumber of elements. After calculating the array factor, the total fieldis obtained by the pattern multiplication rule which is such that thetotal field is the product of the array factor in the field from onesingle element.

E _(total) =E _(single element)×AF

This formula is valid for all arrays consisting of identical elements.The array factor does not depend on the type of elements used, so forcalculating AF it is preferred to use point sources instead of theactual antennas. After calculating the AF, the equation above is used toobtain the total field. Arrays can be 1D (linear), 2D (planar) or 3D. Ina linear array, the elements are placed along the line and in a planarthey are situated in a plane.

Referring now to FIG. 75, there is illustrated in the manner in whichHermite Gaussian beams and Laguerre Gaussian beams will diverge whentransmitted from a phased array of antennas. For the generation ofLaguerre Gaussian beams a circular symmetry over the cross-section ofthe phased antenna array is used, and thus, a circular grid will beutilized. For the generation of Hermite Gaussian beams 7502, arectangular array 7504 of array elements 7506 is utilized. As can beseen with respect to FIG. 75, the Hermite Gaussian waves 7508 provide amore focused beam front then the Laguerre Gaussian waves 7510.

Reduced beam divergence may also be accomplished using a pair of lenses.As illustrated in FIG. 76A, a Gaussian wave 7602 passing through aspiral phase plate 7604 generates an output Laguerre Gaussian wave 7606.The Laguerre Gaussian wave 7606 when passing from a transmitter aperture7608 to a receiver aperture 7610 diverges such that the entire LaguerreGaussian beam does not intersect the receiver aperture 7610. This issuemay be addressed as illustrated in FIG. 76B. As before the Gaussianwaves 7602 pass through the spiral phase plate 7604 generating LaguerreGaussian waves 7606. Prior to passing through the transmitter aperture7608 the Laguerre Gaussian waves 7606 pass through a pair of lenses7614. The pair of lenses 7614 have an effective focal length 7616 thatfocuses the beam 7618 passing through the transmitter aperture 7608. Dueto the focusing lenses 7614, the focused beam 7618 fully intersects thereceiver aperture 7612. By providing the lenses 7614 separated by aneffective focal length 7616, a more focused beam 7618 may be provided atthe receiver aperture 7612 preventing the loss of data within thetransmission of the Laguerre Gaussian wave 7606.

Application of OAM to Optical Communication

Utilization of OAM for optical communications is based on the fact thatcoaxially propagating light beams with different OAM states can beefficiently separated. This is certainly true for orthogonal modes suchas the LG beam. Interestingly, it is also true for general OAM beamswith cylindrical symmetry by relying only on the azimuthal phase.Considering any two OAM beams with an azimuthal index of

1 and

2, respectively:

U ₁(r,θ,z)=A ₁(r,z)exp(il ₁θ)  (12)

where r and z refers to the radial position and propagation distancerespectively, one can quickly conclude that these two beams areorthogonal in the sense that:

$\begin{matrix}{{\int\limits_{0}^{2\pi}{U_{1}U_{2}^{*}d\; \theta}} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} 1_{1}} \neq 1_{2}} \\{A_{1}A_{2}^{*}} & {{{if}\mspace{14mu} 1_{1}} = 1_{2}}\end{matrix} \right.} & (13)\end{matrix}$

There are two different ways to take advantage of the distinctionbetween OAM beams with different

states in communications. In the first approach, N different OAM statescan be encoded as N different data symbols representing “0”, “1”, . . ., “N−1”, respectively. A sequence of OAM states sent by the transmittertherefore represents data information. At the receiver, the data can bedecoded by checking the received OAM state. This approach seems to bemore favorable to the quantum communications community, since OAM couldprovide for the encoding of multiple bits (log 2(N)) per photon due tothe infinitely countable possibilities of the OAM states, and so couldpotentially achieve a higher photon efficiency. The encoding/decoding ofOAM states could also have some potential applications for on-chipinterconnection to increase computing speed or data capacity.

The second approach is to use each OAM beam as a different data carrierin an SDM (Spatial Division Multiplexing) system. For an SDM system, onecould use either a multi-core fiber/free space laser beam array so thatthe data channels in each core/laser beam are spatially separated, oruse a group of orthogonal mode sets to carry different data channels ina multi-mode fiber (MMF) or in free space. Greater than 1 petabit/s datatransmission in a multi-core fiber and up to 6 linearly polarized (LP)modes each with two polarizations in a single core multi-mode fiber hasbeen reported. Similar to the SDM using orthogonal modes, OAM beams withdifferent states can be spatially multiplexed and demultiplexed, therebyproviding independent data carriers in addition to wavelength andpolarization. Ideally, the orthogonality of OAM beams can be maintainedin transmission, which allows all the data channels to be separated andrecovered at the receiver. A typical embodiments of OAM multiplexing isconceptually depicted in FIG. 27. An obvious benefit of OAM multiplexingis the improvement in system spectral efficiency, since the samebandwidth can be reused for additional data channels.

Optical Fiber Communications

The use of orbital angular momentum and multiple layer overlaymodulation processing techniques within an optical communicationsinterface environment as described with respect to FIG. 3 can provide anumber of opportunities within the optical communications environmentfor enabling the use of the greater signal bandwidths provided by theuse of optical orbital angular momentum processing, or multiple layeroverlay modulation techniques alone. FIG. 77 illustrates the generalconfiguration of an optical fiber communication system. The opticalfiber communication system 7700 includes an optical transmitter 7702 andan optical receiver 7704. The transmitter 7702 and receiver 7704communicate over an optical fiber 7706. The transmitter 7702 includesinformation within a light wavelength or wavelengths that is propagatedover the optical fiber 7706 to the optical receiver 7704.

Optical communications network traffic has been steadily increasing by afactor of 100 every decade. The capacity of single mode optical fibershas increased 10,000 times within the last three decades. Historically,the growth in the bandwidth of optical fiber communications has beensustained by information multiplexing techniques using wavelength,amplitude, phase, and polarization of light as a means for encodinginformation. Several major discoveries within the fiber-optics domainhave enabled today's optical networks. An additional discovery was ledby Charles M. Kao's groundbreaking work that recognized glass impuritieswithin an optical fiber as a major signal loss mechanism. Existing glasslosses at the time of his discovery were approximately 200 dB perkilometer at 1 micrometer.

These discoveries gave birth to optical fibers and led to the firstcommercial optical fibers in the 1970s, having an attenuation low enoughfor communication purposes in the range of approximately 20 dBs perkilometer. Referring now to FIGS. 78a-78c , there is more particularlyillustrated the single mode fiber 7802, multicore fibers 7808, andmultimode fibers 7810 described herein above. The multicore fibers 7808consist of multiple cores 7812 included within the cladding 7813 of thefiber. As can be seen in FIG. 78b , there are illustrated a 3 corefiber, 7 core fiber, and 19 core fiber. Multimode fibers 7810 comprisemultimode fibers comprising a few mode fiber 7820 and a multimode fiber7822. Finally, there is illustrated a hollow core fiber 7815 including ahollow core 7814 within the center of the cladding 7816 and sheathing7818. The development of single mode fibers (SMF) such as thatillustrated at 7802 (FIG. 78a ) in the early 1980s reduced pulsedispersion and led to the first fiber-optic based trans-Atlantictelephone cable. This single mode fiber included a single transmissioncore 7804 within an outer sheathing 7806. Development of indium galliumarsenide photodiodes in the early 1990s shifted the focus tonear-infrared wavelengths (1550 NM), were silica had the lowest loss,enabling extended reach of the optical fibers. At roughly the same time,the invention of erbium-doped fiber amplifiers resulted in one of thebiggest leaps in fiber capacity within the history of communication, athousand fold increase in capacity occurred over a 10 year period. Thedevelopment was mainly due to the removed need for expensive repeatersfor signal regeneration, as well as efficient amplification of manywavelengths at the same time, enabling wave division multiplexing (WDM).

Throughout the 2000s, increases in bandwidth capacity came mainly fromintroduction of complex signal modulation formats and coherentdetection, allowing information encoding using the phase of light. Morerecently, polarization division multiplexing (PDM) doubled channelcapacity. Through fiber communication based on SMFs featured tremendousgrowth in the last three decades, recent research has indicated SMFlimitations. Non-linear effects in silica play a significant role inlong range transmission, mainly through the Kerr effect, where apresence of a channel at one wavelength can change the refractive indexof a fiber, causing distortions of other wavelength channels. Morerecently, a spectral efficiency (SE) or bandwidth efficiency, referringto the transmitted information rate over a given bandwidth, has becometheoretically analyzed assuming nonlinear effects in a noisy fiberchannel. This research indicates a specific spectral efficiency limitthat a fiber of a certain length can reach for any signal to noise(SNR). Recently achieved spectral efficiency results indeed show thatthe proximity to the spectral efficiency limit, indicating the need fornew technologies to address the capacity issue in the future.

Among several possible directions for optical communications in thefuture, the introduction of new optical fibers 7706 other than singlemode fibers 7802 has shown promising results. In particular, researchershave focused on spatial dimensions in new fibers, leading to so-calledspace division multiplexing (SDM) where information is transmitted usingcores of multi-core fibers (MCF) 7808 (FIG. 78b ) or mode divisionmultiplexing (MDM) or information is transmitted using modes ofmultimode fibers (MMFs) 7810 (FIG. 78c ). The latest results showspectral efficiency of 91 bits/S/Hz using 12 core multicore fiber 7808for 52 kilometer long fibers and 12 bits/S/Hz using 6 mode multimodefiber 7810 and 112 kilometer long fibers. Somewhat unconventionaltransmissions at 2.08 micrometers have also been demonstrated in two 90meter long photonic crystal fibers, though these fibers had high lossesof 4.5 decibels per kilometer.

While offering promising results, these new types of fibers have theirown limitations. Being noncircularly symmetric structures, multicorefibers are known to require more complex, expensive manufacturing. Onthe other hand, multimode fibers 7810 are easily created using existingtechnologies. However, conventional multimode fibers 7810 are known tosuffer from mode coupling caused by both random perturbations in thefibers and in modal multiplexers/demultiplexers.

Several techniques have been used for mitigating mode coupling. In astrong coupling regime, modal cross talk can be compensated usingcomputationally intensive multi-input multi-output (MIMO) digital signalprocessing (DSP). While MIMO DSP leverages the technique's currentsuccess in wireless networks, the wireless network data rates areseveral orders of magnitude lower than the ones required for opticalnetworks. Furthermore, MIMO DSP complexity inevitably increases with anincreasing number of modes and no MIMO based data transmissiondemonstrations have been demonstrated in real time thus far.Furthermore, unlike wireless communication systems, optical systems arefurther complicated because of fiber's nonlinear effects. In a weakcoupling regime, where cross talk is smaller, methods that also usecomputationally intensive adapted optics, feedback algorithms have beendemonstrated. These methods reverse the effects of mode coupling bysending a desired superposition of modes at the input, so that desiredoutput modes can be obtained. This approach is limited, however, sincemode coupling is a random process that can change on the order of amillisecond in conventional fibers.

Thus, the adaptation of multimode fibers 7810 can be problematic in longhaul systems where the round trip signal propagation delay can be tensof milliseconds. Though 2×56 GB/S transmission at 8 kilometers lengthhas been demonstrated in the case of two higher order modes, none of theadaptive optics MDM methods to date have demonstrated for more than twomodes. Optical fibers act as wave guides for the information carryinglight signals that are transmitted over the fiber. Within an ideal case,optical fibers are 2D, cylindrical wave guides comprising one or severalcores surrounded by a cladding having a slightly lower refractive indexas illustrated in FIGS. 78a-78d . A fiber mode is a solution (aneigenstate) of a wave guide equation describing the field distributionthat propagates within a fiber without changing except for the scalingfactor. All fibers have a limit on the number of modes that they canpropagate, and have both spatial and polarization degrees of freedom.

Single mode fibers (SMFs) 7802 is illustrated in FIG. 78a supportpropagation of two orthogonal polarizations of the fundamental mode only(N=2). For sufficiently large core radius and/or the core claddingdifference, a fiber is multimoded for N >2 as illustrated in FIG. 78c .For optical signals having orbital angular momentums and multilayermodulation schemes applied thereto, multimode fibers 7810 that areweakly guided may be used. Weakly guided fibers have a core claddingrefractive index difference that is very small. Most glass fibersmanufactured today are weakly guided, with the exception of somephotonic crystal fibers and air-core fibers. Fiber guide modes ofmultimode fibers 7810 may be associated in step indexed groups where,within each group, modes typically having similar effective indexes aregrouped together. Within a group, the modes are degenerate. However,these degeneracies can be broken in a certain fiber profile design.

We start by describing translationally invariant waveguide withrefractive index n=n(x, y), with n_(co) being maximum refractive index(“core” of a waveguide), and n_(cl) being refractive index of theuniform cladding, and ρ represents the maximum radius of the refractiveindex n. Due to translational invariance the solutions (or modes) forthis waveguide can be written as:

E _(j)(x,y,z)=e _(j)(x,y)e ^(iβ) ^(j) ^(z),

H _(j)(x,y,z)=h _(j)(x,y)e ^(iβ) ^(j) ^(z),

where β_(j) is the propagation constant of the j-th mode. Vector waveequation for source free Maxwell's equation can be written in this caseas:

(∇² +n ² k ²−β_(j) ²)e _(j)=−(∇_(t) +iβ _(j) {circumflex over (z)})(e_(tj)·∇_(t) ln(n ²))

(∇² +n ² k ²−β_(j) ²)h _(j)=−(∇_(t) ln(n ²))×([(∇]_(t) +iβ _(j){circumflex over (z)})×h _(j))

where k=2π/λ) is the free-space wavenumber, λ is a free-spacewavelength, e_(t)=e_(x){circumflex over (x)}+e_(y)ŷ is a transverse partof the electric field, ∇² is a transverse Laplacian and ∇_(t) transversevector gradient operator. Waveguide polarization properties are builtinto the wave equation through the ∇_(t) ln(n²) terms and ignoring themwould lead to the scalar wave equation, with linearly polarized modes.While previous equations satisfy arbitrary waveguide profile n(x, y), inmost cases of interest, profile height parameter Δ can be consideredsmall Δ<<1, in which case waveguide is said to be weakly guided, or thatweakly guided approximation (WGA) holds. If this is the case, aperturbation theory can be applied to approximate the solutions as:

${E\left( {x,y,z} \right)} = {{{e\left( {x,y} \right)}e^{{i{({\beta + \overset{\sim}{\beta}})}}z}} = {\left( {e_{t} + {\hat{z}e_{z}}} \right)e^{{i{({\beta + \overset{\sim}{\beta}})}}z}}}$${H\left( {x,y,z} \right)} = {{{h\left( {x,y} \right)}r^{{i{({\beta + \overset{\sim}{\beta}})}}z}} = {\left( {h_{t} + {\hat{z}h_{z}}} \right)e^{{i{({\beta + \overset{\sim}{\beta}})}}z}}}$

where subscripts t and z denote transverse and longitudinal componentsrespectively. Longitudinal components can be considered much smaller inWGA and we can approximate (but not neglect) them as:

$e_{z} = {\frac{{i\left( {2\Delta} \right)}^{\frac{1}{2}}}{\; v}\left( {\rho {\nabla_{t}{\cdot e_{t}}}} \right)}$$h_{z} = {\frac{{i\left( {2\Delta} \right)}^{\frac{1}{2}}}{V}\left( {\rho {\nabla_{t}{\cdot h_{t}}}} \right)}$

Where Δ and V are profile height and fiber parameters and transversalcomponents satisfy the simplified wave equation.

(∇² +n ² k ²−β_(j) ²)e _(j)=0

Though WGA simplified the waveguide equation, further simplification canbe obtained by assuming circularly symmetric waveguide (such as idealfiber). If this is the case refractive index that can be written as:

n(r)=n ² _(co)(1 −2f(R)Δ)

where f(R)≥0 is a small arbitrary profile variation.

For a circularly symmetric waveguide, we would have propagationconstants β_(lm) that are classified using azimuthal (l) and radial (m)numbers. Another classification uses effective indices n_(lm) (sometimesnoted as n^(eff) _(lm) or simply n_(eff), that are related topropagation constant as: β_(lm)=kn^(eff)). For the case of l=0, thesolutions can be separated into two classes that have either transverseelectric (T E_(0m)) or transverse magnetic (T M_(0m)) fields (calledmeridional modes). In the case of l≠0, both electric and magnetic fieldhave z-component, and depending on which one is more dominant, so-calledhybrid modes are denoted as: HE_(lm) and EH_(lm).

Polarization correction δβ has different values within the same group ofmodes with the same orbital number (I), even in the circularly symmetricfiber. This is an important observation that led to development of aspecial type of fiber.

In case of a step refractive index, solutions are the Bessel functionsof the first kind, J_(l)(r), in the core region, and modified Besselfunctions of the second kind, K_(l)(r), in the cladding region.

In the case of step-index fiber the groups of modes are almostdegenerate, also meaning that the polarization correction δβ can beconsidered very small. Unlike HE₁₁ modes, higher order modes (HOMs) canhave elaborate polarizations. In the case of circularly symmetric fiber,the odd and even modes (for example HE^(odd) and HE^(even) modes) arealways degenerate (i.e. have equal n_(eff)), regardless of the indexprofile. These modes will be non-degenerate only in the case ofcircularly asymmetric index profiles.

Referring now to FIG. 79, there are illustrated the first six modeswithin a step indexed fiber for the groups L=0 and L=1.

When orbital angular momentums are applied to the light wavelengthwithin an optical transmitter of an optical fiber communication system,the various orbital angular momentums applied to the light wavelengthmay transmit information and be determined within the fiber mode.

Angular momentum density (M) of light in a medium is defined as:

$M = {{\frac{1}{c^{2}}r \times \left( {E \times H} \right)} = {{r \times P} = {\frac{1}{c^{2}}r \times S}}}$

with r as position, E electric field, H magnetic field, P linearmomentum density and S Poynting vector.

The total angular momentum (J), and angular momentum flux (Φ_(M)) can bedefined as:

J=∫∫∫M dV

Φ=∫∫M dA

In order to verify whether certain mode has an OAM let us look at thetime averages of the angular momentum flux Φ_(M):

Φ_(M)

=∫∫

M

dA

as well as the time average of the energy flux:

${\langle\Phi_{M}\rangle} = {\int{\int{\frac{\langle S_{z}\rangle}{c}{dA}}}}$

Because of the symmetry of radial and axial components about the fiberaxis, we note that the integration in equation will leave onlyz-component of the angular momentum density non zero. Hence:

${\langle M\rangle} = {{\langle M\rangle}_{z} = {\frac{1}{c^{2}}r \times {\langle{E \times H}\rangle}_{z}}}$

and knowing (S)=Re{S} and S=½ E×H* leads to:

$S_{\Phi} = {\frac{1}{2}\left( {{{- E_{r}}H_{z}^{*}} + {E_{z}H_{r}^{*}}} \right)}$$S_{z} = {\frac{1}{2}\left( {{E_{x}H_{y}^{*}} - {E_{y}H_{x}^{*}}} \right)}$

Let us now focus on a specific linear combination of the HE_(l+1,m)^(even) and HE_(l+1,m) ^(odd) modes with π/2 phase shift among them:

V _(lm) ⁺ =HE _(l+1,m) ^(even) +iEH _(l+1,m) ^(odd)

The idea for this linear combination comes from observing azimuthaldependence of the HE_(l+1,m) ^(even) and modes comprising cos(φ) and sin(φ). If we denote the electric field of HE_(l+1,m) ^(even) andHE_(l+1,m) ^(odd) modes as e₁ and e₂, respectively, and similarly,denote their magnetic fields as h₁ and h₂, the expression for this newmode can be written as:

e=e ₁ ie ₂,  (2.35)

h=ih ₂.  (2.36)

then we derive:

e_(r) = e^(i(l + 1)ϕ)F_(l)(R)$h_{z} = {e^{{i{({l + 1})}}\phi}n_{co}\mspace{14mu} \left( \frac{\epsilon_{0}}{\mu_{0}} \right)^{\frac{1}{2}}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}\mspace{14mu} G_{l}^{-}}$$e_{z} = {{ie}^{{i{({l + 1})}}\phi}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}G_{l}^{-}}$$h_{r} = {{- {ie}^{{i{({l + 1})}}\phi}}n_{co}\mspace{14mu} \left( \frac{\epsilon_{0}}{\mu_{0}} \right)^{\frac{1}{2}}\mspace{14mu} {F_{l}(R)}}$

Where F_(l)(R) is the Bessel function and

$G_{l}^{\pm} = {\frac{{dF}_{l}}{dR} \pm {\frac{l}{R}F_{l}}}$

We note that all the quantities have e^(i(l+1)φ) dependence thatindicates these modes might have OAM, similarly to the free space case.Therefore the azimuthal and the longitudinal component of the Poyntingvector are:

$S_{\phi} = {{- n_{co}}\mspace{14mu} \left( \frac{\epsilon_{0}}{\mu_{0}} \right)^{\frac{1}{2}}\frac{\left( {2\Delta} \right)^{\frac{1}{2}}}{V}{Re}\left\{ {F_{l}^{*}G_{l}^{-}} \right\}}$$S_{z} = {n_{co}\mspace{14mu} \left( \frac{\epsilon_{0}}{\mu_{0}} \right)^{\frac{1}{2}}\left\lceil F_{l} \right\rceil^{2}}$

The ratio of the angular momentum flux to the energy flux thereforebecomes:

$\frac{\varphi_{M}}{\varphi_{W}} = \frac{l + 1}{\omega}$

We note that in the free-space case, this ratio is similar:

$\frac{\varphi_{M}}{\varphi_{W}} = \frac{\sigma + 1}{\omega}$

where σ represents the polarization of the beam and is bounded to be−1<σ<1. In our case, it can be easily shown that SAM of the V⁺ state, is1, leading to important conclusion that the OAM of the V^(+lm) state isl. Hence, this shows that, in an ideal fiber, OAM mode exists.

Thus, since an orbital angular momentum mode may be detected within theideal fiber, it is possible to encode information using this OAM mode inorder to transmit different types of information having differentorbital angular momentums within the same optical wavelength.

The above description with respect to optical fiber assumed an idealscenario of perfectly symmetrical fibers having no longitudinal changeswithin the fiber profile. Within real world fibers, random perturbationscan induce coupling between spatial and/or polarization modes, causingpropagating fields to evolve randomly through the fiber. The randomperturbations can be divided into two classes, as illustrated in FIG.80. Within the random perturbations 8002, the first class comprisesextrinsic perturbations 8004. Extrinsic perturbations 8004 includestatic and dynamic fluctuations throughout the longitudinal direction ofthe fiber, such as the density and concentration fluctuations natural torandom glassy polymer materials that are included within fibers. Thesecond class includes extrinsic variations 8006 such as microscopicrandom bends caused by stress, diameter variations, and fiber coredefects such as microvoids, cracks, or dust particles.

Mode coupling can be described by field coupling modes which account forcomplex valued modal electric field amplitudes, or by power couplingmodes, which is a simplified description that accounts only for realvalue modal powers. Early multimode fiber systems used incoherent lightemitting diode sources and power coupling models were widely used todescribe several properties including steady state, modal powerdistributions, and fiber impulse responses. While recent multimode fibersystems use coherent sources, power coupling modes are still used todescribe effects such as reduced differential group delays and plasticmultimode fibers.

By contrast, single mode fiber systems have been using laser sources.The study of random birefringence and mode coupling in single modefibers which leads to polarization mode dispersion (PMD), uses fieldcoupling modes which predict the existence of principal states ofpolarization (PSPs). PSPs are polarization states shown to undergominimal dispersion and are used for optical compensation of polarizationmode dispersion in direct detection single mode fiber systems. In recentyears, field coupling modes have been applied to multimode fibers,predicting principal mode which are the basis for optical compensationof modal dispersion in direct detection multimode fiber systems.

Mode coupling can be classified as weak or strong, depending on whetherthe total system length of the optical fiber is comparable to, or muchlonger than, a length scale over which propagating fields remaincorrelated. Depending on the detection format, communication systems canbe divided into direct and coherent detection systems. In directdetection systems, mode coupling must either be avoided by carefuldesign of fibers and modal D (multiplexers) and/or mitigated by adaptiveoptical signal processing. In systems using coherent detection, anylinear cross talk between modes can be compensated by multiple inputmultiple output (MIMO) digital signal processing (DSP), as previouslydiscussed, but DSP complexity increases with an increasing number ofmodes.

Referring now to FIG. 81, there were illustrated the intensity patternsof the first order mode group within a vortex fiber. Arrows 8102 withinthe illustration show the polarization of the electric field within thefiber. The top row illustrates vector modes that are the exact vectorsolutions, and the bottom row shows the resultant, unstable LP11 modescommonly obtained at a fiber output. Specific linear combinations ofpairs of top row modes resulting in the variety of LP11 modes obtainedat the fiber output. Coupled mode 8102 is provided by the coupled pairof mode 8104 and 8106. Coupled mode 8104 is provided by the coupled pairof mode 8104 and mode 8108. Coupled mode 8116 is provided by the coupledpair of mode 8106 and mode 8110, and coupled mode 8118 is provided bythe coupled pair of mode 8108 and mode 8110.

Typically, index separation of two polarizations and single mode fibersis on the order of 10-7. While this small separation lowers the PMD ofthe fiber, external perturbations can easily couple one mode intoanother, and indeed in a single mode fiber, arbitrary polarizations aretypically observed at the output. Simple fiber polarization controllerthat uses stress induced birefringence can be used to achieve anydesired polarization at the output of the fiber.

By the origin, mode coupling can be classified as distributed (caused byrandom perturbations in fibers), or discrete (caused at the modalcouplers and the multiplexers). Most importantly, it has been shown thatsmall, effective index separation among higher order modes is the mainreason for mode coupling and mode instabilities. In particular, thedistributed mode coupling has been shown to be inversely proportional toΔ−P with P greater than 4, depending on coupling conditions. Modeswithin one group are degenerate. For this reason, in most multimodefiber modes that are observed in the fiber output are in fact the linearcombinations of vector modes and are linearly polarized states. Hence,optical angular momentum modes that are the linear combination of the HEeven, odd modes cannot coexist in these fibers due to coupling todegenerate TE01 and TM01 states.

Thus, the combination of the various OAM modes is not likely to generatemodal coupling within the optical systems and by increasing the numberof OAM modes, the reduction in mode coupling is further benefited.

Referring now to FIGS. 82A and 82B, there is illustrated the benefit ofeffective index separation in first order modes. FIG. 82A illustrates atypical step index multimode fiber that does not exhibit effective indexseparation causing mode coupling. The mode TM₀₁HE^(even) ₂₁, modeHE^(odd) ₂₁, and mode TE₀₁ have little effective index separation, andthese modes would be coupled together. Mode HE^(x,1) ₁₁ has an effectiveindex separation such that this mode is not coupled with these othermodes.

This can be compared with the same modes in FIG. 82B. In this case,there is an effective separation 8202 between the TM₀₁ mode and theHE^(even) ₂₁ mode and the TE₀₁ mode and the HE^(odd) ₂₁ mode. Thiseffective separation causes no mode coupling between these mode levelsin a similar manner that was done in the same modes in FIG. 82A.

In addition to effective index separation, mode coupling also depends onthe strength of perturbation. An increase in the cladding diameter of anoptical fiber can reduce the bend induced perturbations in the fiber.Special fiber design that includes the trench region can achieveso-called bend insensitivity, which is predominant in fiber to the home.Fiber design that demonstrates reduced bends and sensitivity of higherorder Bessel modes for high power lasers have been demonstrated. Mostimportant, a special fiber design can remove the degeneracy of the firstorder mode, thus reducing the mode coupling and enabling the OAM modesto propagate within these fibers.

Topological charge may be multiplexed to the wave length for eitherlinear or circular polarization. In the case of linear polarizations,topological charge would be multiplexed on vertical and horizontalpolarization. In case of circular polarization, topological charge wouldbe multiplexed on left hand and right hand circular polarization.

The topological charges can be created using Spiral Phase Plates (SPPs)such as that illustrated in FIG. 11e , phase mask holograms or a SpatialLight Modulator (SLM) by adjusting the voltages on SLM which createsproperly varying index of refraction resulting in twisting of the beamwith a specific topological charge. Different topological charges can becreated and muxed together and de-muxed to separate charges. Whensignals are muxed together, multiple signals having different orthogonalfunctions or helicities applied thereto are located in a same signal.The muxed signals are spatially combined in a same signal.

As Spiral Phase plates can transform a plane wave (1=0) to a twistedwave of a specific helicity (i.e. 1=+1), Quarter Wave Plates (QWP) cantransform a linear polarization (s=0) to circular polarization (i.e.s=+1).

Cross talk and multipath interference can be reduced usingMultiple-Input-Multiple-Output (MIMO).

Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system).

Optical Fiber Communications Using OAM Multiplexing

OAM multiplexing may be implemented in fiber communications. OAM modesare essentially a group of higher order modes defined on a differentbasis as compared to other forms of modes in fiber, such as “linearlypolarized” (LP) modes and fiber vector modes. In principle each of themode sets form an orthogonal mode basis spanning the spatial domain, andmay be used to transmit different data channels. Both LP modes and OAMmodes face challenges of mode coupling when propagating in a fiber, andmay also cause channel crosstalk problems.

In general, two approaches may be involved in fiber transmission usingOAM multiplexing. The first approach is to implement OAM transmission ina regular few mode fiber such as that illustrated in FIG. 78. As is thecase of SDM using LP modes, MIMO DSP is generally required to equalizethe channel interface. The second approach is to utilize a speciallydesigned vortex fiber that suffers from less mode coupling, and DSPequalization can therefore be saved for a certain distance oftransmission.

OAM Transmission in Regular Few Mode Fiber

In a regular few mode fiber, each OAM mode represents approximately alinear combination of the true fiber modes (the solution to the waveequation in fiber). For example, as illustrated in FIG. 83, a linearlypolarized OAM beam 8302 with

=+1 comprises the components of Eigen modes including TE₀₁, TM₀₁ andHE₂₁. Due to the perturbations or other non-idealities, OAM modes thatare launched into a few mode fiber (FMF) may quickly coupled to eachother, most likely manifesting in a group of LP modes at the fiberoutput. The mutual mode coupling in fiber may lead to inter-channelcrosstalk and eventually failure of the transmission. One possiblesolution for the mode coupling effects is to use MIMO DSP in combinationwith coherent detection.

Referring now to FIG. 84, there is illustrated a demonstration of thetransmission of four OAM beams (

=+1 and −1 each with 2 orthogonal polarization states), each carrying 20Gbit/s QPSK data, in an approximately 5 kilometer regular FMF (few modefiber) 8404. Four data channels 8402 (2 with x-pol and 2 with y-pol)were converted to pol-muxed OAM beams with

=+1 and −1 using an inverse mode sorter 8406. The pol-muxed to OAM beams8408 (four in total) are coupled into the FMF 8404 for propagation. Δtthe fiber output, the received modes were decomposed onto an OAM basis (

=+1 and −1) using a mode sorter 8410. In each of the two OAM componentsof light were coupled onto a fiber-based PBS for polarizationdemultiplexing. Each output 8412 is detected by a photodiode, followedby ADC (analog-to-digital converter) and off-line processing. Tomitigate the inter-channel interference, a constant modulus algorithm isused to blindly estimate the channel crosstalk and compensate for theinter-channel interference using linear equalization. Eventually, theQPSK data carried on each OAM beam is recovered with the assistance of aMIMO DSP as illustrated in FIGS. 85A and 85B.

OAM Transmission in a Vortex Fiber

A key challenge for OAM multiplexing in conventional fibers is thatdifferent OAM modes tend to couple to each other during thetransmission. The major reason for this is that in a conventional fiberOAM modes have a relatively small effective refractive index difference(Δn_(eff)). Stably transmitting an OAM mode in fiber requires somemodifications of the fiber. One manner for stably transmitting OAM modesuses a vortex fiber such as that illustrated in FIG. 86. A vortex fiber8602 is a specially designed a few mode fiber including an additionalhigh index ring 8604 around the fiber core 8606. The design increasesthe effective index differences of modes and therefore reduces themutual mode coupling.

Using this vortex fiber 8602, two OAM modes with

=+1 and −1 and two polarizations multiplexed fundamental modes weretransmitted together for 1.1 km. The measured mode cross talk betweentwo OAM modes was approximately −20 dB. These four distinct modes wereused to each carried a 100 Gbuad QPSK signal at the same wavelength andsimultaneously propagate in the vortex fiber. After the modedemultiplexing, all data was recovered with a power penalty ofapproximately 4.1 dB, which could be attributed to the multipath effectsand mode cross talk. In a further example, WDM was added to furtherextend the capacity of a vortex fiber transmission system. A 20 channelfiber link using to OAM modes and 10 WDM channels (from 1546.642 nm to1553.88 nm), each channel sending 80 Gb/s 16-QAM signal wasdemonstrated, resulting in a total transmission capacity of 1.2 Tb/sunder the FEC limit.

There are additional innovative efforts being made to design andfabricate fibers that are more suitable for OAM multiplexing. A recentlyreported air-core fiber has been demonstrated to further increase therefractive index difference of eigenmodes such that the fiber is able tostably transmit 12 OAM states (

=±7, ±8 and ±9, each with two orthogonal polarizations) for 2 m. A fewmode fibers having an inverse parabolic graded index profile in whichpropagating 8 OAM orders (

=±1 and ±2, each with two orthogonal polarizations) has beendemonstrated over 1.1 km. The same group recently presented a newerversion of an air core fiber, whereby the supported OAM states wasincreased to 16. One possible design that can further increase thesupported OAM modes and a fiber is to use multiple high contrast indexedring core structure which is indicated a good potential for OAMmultiplexing for fiber communications.

RF Communications with OAM

As a general property of electromagnetic waves, OAM can also be carriedon other ways with either a shorter wavelength (e.g., x-ray), or alonger wavelength (millimeter waves and terahertz waves) than an opticalbeam. Focusing on the RF waves, OAM beams at 90 GHz were initiallygenerated using a spiral phase plate made of Teflon. Differentapproaches, such as a phase array antenna and a helicoidal parabolicantenna have also been proposed. RF OAM beams have been used as datacarriers for RF communications. A Gaussian beam and an OAM beam with

=+1 at approximately 2.4 GHz have been transmitted by a Yagi-Uda antennaand a spiral parabolic antenna, respectively, which are placed inparallel. These two beams were distinguished by the differential outputof a pair of antennas at the receiver side. The number of channels wasincreased to three (carried on OAM beams with

=−1, 0 and +1) using a similar apparatus to send approximately 11 Mb/ssignal at approximately 17 GHz carrier. Note that in these twodemonstrations different OAM beams propagate along different spatialaxes. There are some potential benefits if all of the OAM beams areactually multiplexed and propagated through the same aperture. In arecent demonstration eight polarization multiplexed (pol-muxed) RF OAMbeams (for OAM beams on each of two orthogonal polarizations) ourcoaxially propagated through a 2.5 m link.

The herein described RF techniques have application in a wide variety ofRF environments. These include RF Point to Point/Multipointapplications, RF Point to Point Backhaul applications, RF Point to PointFronthaul applications (these provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and future cloudifiedHetNet), RF Satellite applications, RF Wifi (LAN) applications, RFBluetooth (PAN) applications, RF personal device cable replacementapplications, RF Radar applications and RF electromagnet tagapplications. The techniques could also be used in a RF and FSO hybridsystem that can provide communications in an RF mode or an FSO modedepending on which mode of operation is providing the most optimal orcost effective communications link at a particular point in time.

The four different OAM beams with

=−3, −1, +1 and +3 on each of 2 orthogonal polarizations are generatedusing customized spiral phase plates specifically for millimeter wave at28 GHz. The observed intensity profile for each of the beams and theirinterferograms are shown in FIG. 87. These OAM beams were coaxiallymultiplexed using designed beam splitters. After propagation, the OAMchannels were multiplexed using an inverse spiral phase plate and aspatial filter (the receiver antenna). The measured crosstalk it 28 GHzfor each of the demultiplexed channels is shown in Table 8. It can beseen that the cross talk is low enough for 16-QAM data transmissionwithout the assistance of extra DSPs to reduce the channel interference.

TABLE 8 Crosstalk of the OAM channels measured at f = 28 GHz (CW)

 = −3

 = −1

 = +1

 = +3 Single-pol (Y-pol) −25 dB   −23 dB   −25 dB −26 dB Dual-pol(X-pol) −17 dB −16.5 dB −18.1 dB −19 dB Dual-pol (Y-pol) −18 dB −16.5 dB−16.5 dB −24 dB

Considering that each beam carries a 1 Gbaud 16-QAM signal, a total linkcapacity of 32 Gb/s at a single carrier frequency of 28 GHz and aspectral efficiency of 16 Gb/s/Hz may be achieved. In addition, an RFOAM beam demultiplexer (“mode sorter”) was also customize for a 28 GHzcarrier and is implemented in such a link to simultaneously separatemultiple OAM beams. Simultaneously demultiplexing for OAM beams at thesingle polarization has been demonstrated with a cross talk of less than−14 dB. The cross talk is likely to be further reduced by optimizing thedesign parameters.

Free Space Communications

An additional configuration in which the optical angular momentumprocessing and multi-layer overlay modulation technique described hereinabove may prove useful within the optical network framework is use withfree-space optics communications. Free-space optics systems provide anumber of advantages over traditional UHF RF based systems from improvedisolation between the systems, the size and the cost of thereceivers/transmitters, lack of RF licensing laws, and by combiningspace, lighting, and communication into the same system. Referring nowto FIG. 88 there is illustrated an example of the operation of afree-space communication system. The free-space communication systemutilizes a free-space optics transmitter 8802 that transmits a lightbeam 8804 to a free-space optics receiver 8806. The major differencebetween a fiber-optic network and a free-space optic network is that theinformation beam is transmitted through free space rather than over afiber-optic cable. This causes a number of link difficulties, which willbe more fully discussed herein below. Free-space optics is a line ofsight technology that uses the invisible beams of light to provideoptical bandwidth connections that can send and receive up to 2.5 Gbpsof data, voice, and video communications between a transmitter 8802 anda receiver 8806. Free-space optics uses the same concepts asfiber-optics, except without the use of a fiber-optic cable. Free-spaceoptics systems provide the light beam 8804 within the infrared (IR)spectrum, which is at the low end of the light spectrum. Specifically,the optical signal is in the range of 300 Gigahertz to 1 Terahertz interms of wavelength.

Presently existing free-space optics systems can provide data rates ofup to 10 Gigabits per second at a distance of up to 2.5 kilometers. Inouter space, the communications range of free space opticalcommunications is currently on the order of several thousand kilometers,but has the potential to bridge interplanetary distances of millions ofkilometers, using optical telescopes as beam expanders. In January of2013, NASA used lasers to beam an image of the Mona Lisa to the LunarReconnaissance Orbiter roughly 240,000 miles away. To compensate foratmospheric interference, an error correction code algorithm, similar tothat used within compact discs, was implemented.

The distance records for optical communications involve detection andemission of laser light by space probes. A two-way distance record forcommunication was established by the Mercury Laser Altimeter instrumentaboard the MESSENGER spacecraft. This infrared diode neodymium laser,designed as a laser altimeter for a Mercury Orbiter mission, was able tocommunicate across a distance of roughly 15,000,000 miles (24,000,000kilometers) as the craft neared Earth on a fly by in May of 2005. Theprevious record had been set with a one-way detection of laser lightfrom Earth by the Galileo Probe as two ground based lasers were seenfrom 6,000,000 kilometers by the outbound probe in 1992. Researchersused a white LED based space lighting system for indoor local areanetwork communications.

Referring now to FIG. 89, there is illustrated a block diagram of afree-space optics system using orbital angular momentum and multileveloverlay modulation according to the present disclosure. The OAM twistedsignals, in addition to being transmitted over fiber, may also betransmitted using free optics. In this case, the transmission signalsare generated within transmission circuitry 8902 at each of the FSOtransceivers 8904. Free-space optics technology is based on theconnectivity between the FSO based optical wireless units, eachconsisting of an optical transceiver 8904 with a transmitter 8902 and areceiver 8906 to provide full duplex open pair and bidirectional closedpairing capability. Each optical wireless transceiver unit 8904additionally includes an optical source 8908 plus a lens or telescope8910 for transmitting light through the atmosphere to another lens 8910receiving the information. At this point, the receiving lens ortelescope 8910 connects to a high sensitivity receiver 8906 via opticalfiber 8912. The transmitting transceiver 8904 a and the receivingtransceiver 8904 b have to have line of sight to each other. Trees,buildings, animals, and atmospheric conditions all can hinder the lineof sight needed for this communications medium. Since line of sight isso critical, some systems make use of beam divergence or a diffused beamapproach, which involves a large field of view that toleratessubstantial line of sight interference without significant impact onoverall signal quality. The system may also be equipped with autotracking mechanism 8914 that maintains a tightly focused beam on thereceiving transceiver 3404 b, even when the transceivers are mounted ontall buildings or other structures that sway.

The modulated light source used with optical source 8908 is typically alaser or light emitting diode (LED) providing the transmitted opticalsignal that determines all the transmitter capabilities of the system.Only the detector sensitivity within the receiver 8906 plays an equallyimportant role in total system performance. For telecommunicationspurposes, only lasers that are capable of being modulated at 20 Megabitsper second to 2.5 Gigabits per second can meet current marketplacedemands. Additionally, how the device is modulated and how muchmodulated power is produced are both important to the selection of thedevice. Lasers in the 780-850 nm and 1520-1600 nm spectral bands meetfrequency requirements.

Commercially available FSO systems operate in the near IR wavelengthrange between 750 and 1600 nm, with one or two systems being developedto operate at the IR wavelength of 10,000 nm. The physics andtransmissions properties of optical energy as it travels through theatmosphere are similar throughout the visible and near IR wavelengthrange, but several factors that influence which wavelengths are chosenfor a particular system.

The atmosphere is considered to be highly transparent in the visible andnear IR wavelength. However, certain wavelengths or wavelength bands canexperience severe absorption. In the near IR wavelength, absorptionoccurs primarily in response to water particles (i.e., moisture) whichare an inherent part of the atmosphere, even under clear weatherconditions. There are several transmission windows that are nearlytransparent (i.e., have an attenuation of less than 0.2 dB perkilometer) within the 700-10,000 nm wavelength range. These wavelengthsare located around specific center wavelengths, with the majority offree-space optics systems designed to operate in the windows of 780-850nm and 1520-1600 nm.

Wavelengths in the 780-850 nm range are suitable for free-space opticsoperation and higher power laser sources may operate in this range. At780 nm, inexpensive CD lasers may be used, but the average lifespan ofthese lasers can be an issue. These issues may be addressed by runningthe lasers at a fraction of their maximum rated output power which willgreatly increase their lifespan. At around 850 nm, the optical source8908 may comprise an inexpensive, high performance transmitter anddetector components that are readily available and commonly used innetwork transmission equipment. Highly sensitive silicon (SI) avalanchephotodiodes (APD) detector technology and advanced vertical cavityemitting laser may be utilized within the optical source 8908.

VCSEL technology may be used for operation in the 780 to 850 nm range.Possible disadvantage of this technology include beam detection throughthe use of a night vision scope, although it is still not possible todemodulate a perceived light beam using this technique.

Wavelengths in the 1520-1600 nm range are well-suited for free-spacetransmission, and high quality transmitter and detector components arereadily available for use within the optical source block 8908. Thecombination of low attenuation and high component availability withinthis wavelength range makes the development of wavelength divisionmultiplexing (WDM) free-space optics systems feasible. However,components are generally more expensive and detectors are typically lesssensitive and have a smaller receive surface area when compared withsilicon avalanche photodiode detectors that operator at the 850 nmwavelength. These wavelengths are compatible with erbium-doped fiberamplifier technology, which is important for high power (greater than500 milliwatt) and high data rate (greater than 2.5 Gigabytes persecond) systems. Fifty to 65 times as much power can be transmitted atthe 1520-1600 nm wavelength than can be transmitted at the 780-850 nmwavelength for the same eye safety classification. Disadvantages ofthese wavelengths include the inability to detect a beam with a nightvision scope. The night vision scope is one technique that may be usedfor aligning the beam through the alignment circuitry 8914. Class 1lasers are safe under reasonably foreseeable operating conditionsincluding the use of optical instruments for intrabeam viewing. Class 1systems can be installed at any location without restriction.

Another potential optical source 8908 comprised Class 1M lasers. Class1M laser systems operate in the wavelength range from 302.5 to 4000 nm,which is safe under reasonably foreseeable conditions, but may behazardous if the user employs optical instruments within some portion ofthe beam path. As a result, Class 1M systems should only be installed inlocations where the unsafe use of optical aids can be prevented.Examples of various characteristics of both Class 1 and Class 1M lasersthat may be used for the optical source 4708 are illustrated in Table 9below.

TABLE 9 Laser Power Aperture Size Distance Power Density Classification(mW) (mm) (m) (mW/cm²) 850-nm Wavelength Class 1 0.78 7 14 2.03 50 20000.04 Class 1M 0.78 7 100 2.03 500 7 14 1299.88 50 2000 25.48 1550-nmWavelength Class 1 10 7 14 26.00 25 2000 2.04 Class 1M 10 3.5 100 103.99500 7 14 1299.88 25 2000 101.91

The 10,000 nm wavelength is relatively new to the commercial free spaceoptic arena and is being developed because of better fog transmissioncapabilities. There is presently considerable debate regarding thesecharacteristics because they are heavily dependent upon fog type andduration. Few components are available at the 10,000 nm wavelength, asit is normally not used within telecommunications equipment.Additionally, 10,000 nm energy does not penetrate glass, so it isill-suited to behind window deployment.

Within these wavelength windows, FSO systems should have the followingcharacteristics. The system should have the ability to operate at higherpower levels, which is important for longer distance FSO systemtransmissions. The system should have the ability to provide high speedmodulation, which is important for high speed FSO systems. The systemshould provide a small footprint and low power consumption, which isimportant for overall system design and maintenance. The system shouldhave the ability to operate over a wide temperature range without majorperformance degradations such that the systems may prove useful foroutdoor systems. Additionally, the mean time between failures shouldexceed 10 years. Presently existing FSO systems generally use VCSELS foroperation in the shorter IR wavelength range, and Fabry-Pérot ordistributed feedback lasers for operation in the longer IR wavelengthrange. Several other laser types are suitable for high performance FSOsystems.

A free-space optics system using orbital angular momentum processing andmulti-layer overlay modulation would provide a number of advantages. Thesystem would be very convenient. Free-space optics provides a wirelesssolution to a last-mile connection, or a connection between twobuildings. There is no necessity to dig or bury fiber cable. Free-spaceoptics also requires no RF license. The system is upgradable and itsopen interfaces support equipment from a variety of vendors. The systemcan be deployed behind windows, eliminating the need for costly rooftopright. It is also immune to radiofrequency interference or saturation.The system is also fairly speedy. The system provides 2.5 Gigabits persecond of data throughput. This provides ample bandwidth to transferfiles between two sites. With the growth in the size of files,free-space optics provides the necessary bandwidth to transfer thesefiles efficiently.

Free-space optics also provides a secure wireless solution. The laserbeam cannot be detected with a spectral analyzer or RF meter. The beamis invisible, which makes it difficult to find. The laser beam that isused to transmit and receive the data is very narrow. This means that itis almost impossible to intercept the data being transmitted. One wouldhave to be within the line of sight between the receiver and thetransmitter in order to be able to accomplish this feat. If this occurs,this would alert the receiving site that a connection has been lost.Thus, minimal security upgrades would be required for a free-spaceoptics system.

However, there are several weaknesses with free-space optics systems.The distance of a free-space optics system is very limited. Currentlyoperating distances are approximately within 2 kilometers. Although thisis a powerful system with great throughput, the limitation of distanceis a big deterrent for full-scale implementation. Additionally, allsystems require line of sight be maintained at all times duringtransmission. Any obstacle, be it environmental or animals can hinderthe transmission. Free-space optic technology must be designed to combatchanges in the atmosphere which can affect free-space optic systemperformance capacity.

Something that may affect a free-space optics system is fog. Dense fogis a primary challenge to the operation of free-space optics systems.Rain and snow have little effect on free-space optics technology, butfog is different. Fog is a vapor composed of water droplets which areonly a few hundred microns in diameter, but can modify lightcharacteristics or completely hinder the passage of light through acombination of absorption, scattering, and reflection. The primaryanswer to counter fog when deploying free-space optic based wirelessproducts is through a network design that shortens FSO linked distancesand adds network redundancies.

Absorption is another problem. Absorption occurs when suspended watermolecules in the terrestrial atmosphere extinguish photons. This causesa decrease in the power density (attenuation) of the free space opticsbeam and directly affects the availability of the system. Absorptionoccurs more readily at some wavelengths than others. However, the use ofappropriate power based on atmospheric conditions and the use of spatialdiversity (multiple beams within an FSO based unit), helps maintain therequired level of network availability.

Solar interference is also a problem. Free-space optics systems use ahigh sensitivity receiver in combination with a larger aperture lens. Asa result, natural background light can potentially interfere withfree-space optics signal reception. This is especially the case with thehigh levels of background radiation associated with intense sunlight. Insome instances, direct sunlight may case link outages for periods ofseveral minutes when the sun is within the receiver's field of vision.However, the times when the receiver is most susceptible to the effectsof direct solar illumination can be easily predicted. When directexposure of the equipment cannot be avoided, the narrowing of receiverfield of vision and/or using narrow bandwidth light filters can improvesystem performance. Interference caused by sunlight reflecting off of aglass surface is also possible.

Scattering issues may also affect connection availability. Scattering iscaused when the wavelength collides with the scatterer. The physicalsize of the scatterer determines the type of scattering. When thescatterer is smaller than the wavelength, this is known as Rayleighscattering. When a scatterer is of comparable size to the wavelengths,this is known as Mie scattering. When the scattering is much larger thanthe wavelength, this is known as non-selective scattering. Inscattering, unlike absorption, there is no loss of energy, only adirectional redistribution of energy that may have significant reductionin beam intensity over longer distances.

Physical obstructions such as flying birds or construction cranes canalso temporarily block a single beam free space optics system, but thistends to cause only short interruptions. Transmissions are easily andautomatically resumed when the obstacle moves. Optical wireless productsuse multibeams (spatial diversity) to address temporary abstractions aswell as other atmospheric conditions, to provide for greateravailability.

The movement of buildings can upset receiver and transmitter alignment.Free-space optics based optical wireless offerings use divergent beamsto maintain connectivity. When combined with tracking mechanisms,multiple beam FSO based systems provide even greater performance andenhanced installation simplicity.

Scintillation is caused by heated air rising from the Earth or man-madedevices such as heating ducts that create temperature variations amongdifferent pockets of air. This can cause fluctuations in signalamplitude, which leads to “image dancing” at the free-space optics basedreceiver end. The effects of this scintillation are called “refractiveturbulence.” This causes primarily two effects on the optical beams.Beam wander is caused by the turbulent eddies that are no larger thanthe beam. Beam spreading is the spread of an optical beam as itpropagates through the atmosphere.

Referring now to FIGS. 90A-90D, in order to achieve higher data capacitywithin optical links, an additional degree of freedom from multiplexingmultiple data channels must be exploited. Moreover, the ability to usetwo different orthogonal multiplexing techniques together has thepotential to dramatically enhance system performance and increasedbandwidth.

One multiplexing technique which may exploit the possibilities is modedivision multiplexing (MDM) using orbital angular momentum (OAM). OAMmode refers to laser beams within a free-space optical system orfiber-optic system that have a phase term of e^(ilφ) in their wavefronts, in which φ is the azimuth angle and l determines the OAM value(topological charge). In general, OAM modes have a “donut-like” ringshaped intensity distribution. Multiple spatial collocated laser beams,which carry different OAM values, are orthogonal to each other and canbe used to transmit multiple independent data channels on the samewavelength. Consequently, the system capacity and spectral efficiency interms of bits/S/Hz can be dramatically increased. Free-spacecommunications links using OAM may support 100 Tbits/capacity. Varioustechniques for implementing this as illustrated in FIGS. 90A-90D includea combination of multiple beams 9002 having multiple different OAMvalues 9004 on each wavelength. Thus, beam 9002 includes OAM values,OAM1 and OAM4. Beam 9006 includes OAM value 2 and OAM value 5. Finally,beam 9008 includes OAM3 value and OAM6 value. Referring now to FIG. 90B,there is illustrated a single beam wavelength 9010 using a first groupof OAM values 9012 having both a positive OAM value 9012 and a negativeOAM value 9014. Similarly, OAM2 value may have a positive value 9016 anda negative value 9018 on the same wavelength 9010. While mode divisionmultiplexing of OAM modes is described above, other orthogonal functionsmay be used with mode division multiplexing such as Laguerre Gaussianfunctions, Hermite Gaussian functions, Jacobi functions, Gegenbauerfunctions, Legendre functions, Chebyshev functions or Ince-Gaussianfunctions.

FIG. 90C illustrates the use of a wavelength 9020 having polarizationmultiplexing of OAM value. The wavelength 9020 can have multiple OAMvalues 9022 multiplexed thereon. The number of available channels can befurther increased by applying left or right handed polarization to theOAM values. Finally, FIG. 90D illustrates two groups of concentric rings9060, 9062 for a wavelength having multiple OAM values.

Wavelength distribution multiplexing (WDM) has been widely used toimprove the optical communication capacity within both fiber-opticsystems and free-space communication system. OAM mode/mode divisionmultiplexing and WDM are mutually orthogonal such that they can becombined to achieve a dramatic increase in system capacity. Referringnow to FIG. 91, there is illustrated a scenario where each WDM channel9102 contains many orthogonal OAM beam 9104. Thus, using a combinationof orbital angular momentum with wave division multiplexing, asignificant enhancement in communication link to capacity may beachieved. By further combining polarization multiplexing with acombination of MDM and WDM even further increased in bandwidth capacitymay be achieved from the +/− polarization values being added to the modeand wavelength multiplexing.

Current optical communication architectures have considerable routingchallenges. A routing protocol for use with free-space optic system musttake into account the line of sight requirements for opticalcommunications within a free-space optics system. Thus, a free-spaceoptics network must be modeled as a directed hierarchical random sectorgeometric graph in which sensors route their data via multi-hop paths toa base station through a cluster head. This is a new efficient routingalgorithm for local neighborhood discovery and a base station uplink anddownlink discovery algorithm. The routing protocol requires orderOlog(n) storage at each node versus order O(n) used within currenttechniques and architectures.

Current routing protocols are based on link state, distance vectors,path vectors, or source routing, and they differ from the new routingtechnique in significant manners. First, current techniques assume thata fraction of the links are bidirectional. This is not true within afree-space optic network in which all links are unidirectional. Second,many current protocols are designed for ad hoc networks in which therouting protocol is designed to support multi-hop communications betweenany pair of nodes. The goal of the sensor network is to route sensorreadings to the base station. Therefore, the dominant traffic patternsare different from those in an ad hoc network. In a sensor network, nodeto base stations, base station to nodes, and local neighborhoodcommunication are mostly used.

Recent studies have considered the effect of unidirectional links andreport that as many as 5 percent to 10 percent of links and wireless adhoc networks are unidirectional due to various factors. Routingprotocols such as DSDV and AODV use a reverse path technique, implicitlyignoring such unidirectional links and are therefore not relevant inthis scenario. Other protocols such as DSR, ZRP, or ZRL have beendesigned or modified to accommodate unidirectionality by detectingunidirectional links and then providing bidirectional abstraction forsuch links. Referring now to FIG. 92, the simplest and most efficientsolution for dealing with unidirectionality is tunneling, in whichbidirectionality is emulated for a unidirectional link by usingbidirectional links on a reverse back channel to establish the tunnel.Tunneling also prevents implosion of acknowledgement packets and loopingby simply pressing link layer acknowledgements for tunneled packetsreceived on a unidirectional link. Tunneling, however, works well inmostly bidirectional networks with few unidirectional links.

Within a network using only unidirectional links such as a free-spaceoptical network, systems such as that illustrated in FIGS. 92 and 93would be more applicable. Nodes within a unidirectional network utilizea directional transmit 9202 transmitting from the node 9200 in a single,defined direction. Additionally, each node 9200 includes anomnidirectional receiver 9204 which can receive a signal coming to thenode in any direction. Also, as discussed here and above, the node 9200would also include a 0 log(n) storage 9206. Thus, each node 9200 provideonly unidirectional communications links. Thus, a series of nodes 9200as illustrated in FIG. 93 may unidirectionally communicate with anyother node 9200 and forward communication from one desk location toanother through a sequence of interconnected nodes.

Topological charge may be multiplexed to the wave length for eitherlinear or circular polarization. In the case of linear polarizations,topological charge would be multiplexed on vertical and horizontalpolarization. In case of circular polarization, topological charge wouldbe multiplexed on left hand and right hand circular polarizations.

The topological charges can be created using Spiral Phase Plates (SPPs)such as that illustrated in FIG. 12E, phase mask holograms or a SpatialLight Modulator (SLM) by adjusting the voltages on SLM which createsproperly varying index of refraction resulting in twisting of the beamwith a specific topological charge. Different topological charges can becreated and muxed together and de-muxed to separate charges.

As Spiral Phase plates can transform a plane wave (1=0) to a twistedwave of a specific helicity (i.e. 1=+1), Quarter Wave Plates (QWP) cantransform a linear polarization (s=0) to circular polarization (i.e.s=+1).

Cross talk and multipath interference can be reduced usingMultiple-Input-Multiple-Output (MIMO).

Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system).

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

In a further embodiment illustrated in FIG. 94, both RF signals and freespace optics may be implemented within a dual RF and free space opticsmechanism 9402. The dual RF and free space optics mechanism 9402 includea free space optics projection portion 9404 that transmits a light wavehaving an orbital angular momentum applied thereto with multileveloverlay modulation and a RF portion 9406 including circuitry necessaryfor transmitting information with orbital angular momentum andmultilayer overlay on an RF signal 9410. The dual RF and free spaceoptics mechanism 9402 may be multiplexed in real time between the freespace optics signal 9408 and the RF signal 9410 depending upon operatingconditions. In some situations, the free space optics signal 9408 wouldbe most appropriate for transmitting the data. In other situations, thefree space optics signal 9408 would not be available and the RF signal9410 would be most appropriate for transmitting data. The dual RF andfree space optics mechanism 9402 may multiplex in real time betweenthese two signals based upon the available operating conditions.

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

Quantum Communication Using OAM

OAM has also received increasing interest for its potential role in thedevelopment of secure quantum communications that are based on thefundamental laws of quantum mechanics (i.e., quantum no cloningtheorem). One of the examples is high dimensional quantum keydistribution (QKD) QKD systems have conventionally utilized thepolarization or phase of light for encoding. The original proposal forQKD (i.e., the BB 84 protocol of Bennett and Brassard) encodesinformation on the polarization states and so only allow one bit ofinformation to be impressed onto each photon. The benefit of using OAMis that OAM states reside in an infinite dimensional Hilbert space,implying the possibility of encoding multiple bits of information on anindividual photon. Similar to the use of OAM multiplexing in classicaloptical communications, the secure key rate can be further increasedsimultaneous encoding of information in different domains is implementedthrough making use of high dimensional entanglement. The addition to theadvantages of a large alphabet for information encoding, the security ofkeys generated by an OAM-based QKD system have been shown to be improveddue to the use of a large Hilbert space, which indicates increaserobustness of the QKD system against eavesdropping.

FIG. 95 illustrates a seven dimensional QKD link based on OAM encoding.FIG. 96 shows the two complementary seven dimensional bases used forinformation encoding. Recent QKD systems have been demonstrated tooperate at a secure key rate of up to 1 Mb/s. However, in order tosupport an OAM-based QKD system with a higher secure key rate, thedevelopment of a OAM generation methods with speeds higher than MHzwould be required. Another challenge arises from the efficiency inchsorting single photons in the OAM basis, although the current OAMsorting approach allows an OAM separation efficiency of greater than92%. Additionally, adverse channel conditions pose a critical challenge.For a free space QKD system employing OAM states, atmospheric turbulencethat distorts the phase front of an OAM state may significantly degradethe information content of the transmitted OAM light field.

Quantum Key Distribution

Referring now to FIG. 97, there is illustrated a further improvement ofa system utilizing orbital angular momentum processing, LaguerreGaussian processing, Hermite Gaussian processing or processing using anyorthogonal functions. In the illustration of FIG. 97, a transmitter 9702and receiver 9704 are interconnected over an optical link 9706. Theoptical link 9706 may comprise a fiber-optic link or a free-space opticlink as described herein above. The transmitter receives a data stream9708 that is processed via orbital angular momentum processing circuitry9710. The orbital angular momentum processing circuitry 9710 provideorbital angular momentum twist to various signals on separate channelsas described herein above. In some embodiments, the orbital angularmomentum processing circuitry may further provide multi-layer overlaymodulation to the signal channels in order to further increase systembandwidth.

The OAM processed signals are provided to quantum key distributionprocessing circuitry 9712. The quantum key distribution processingcircuitry 9712 utilizes the principals of quantum key distribution aswill be more fully described herein below to enable encryption of thesignal being transmitted over the optical link 9706 to the receiver9704. The received signals are processed within the receiver 9704 usingthe quantum key distribution processing circuitry 9714. The quantum keydistribution processing circuitry 9714 decrypts the received signalsusing the quantum key distribution processing as will be more fullydescribed herein below. The decrypted signals are provided to orbitalangular momentum processing circuitry 9716 which removes any orbitalangular momentum twist from the signals to generate the plurality ofoutput signals 9718. As mentioned previously, the orbital angularmomentum processing circuitry 9716 may also demodulate the signals usingmultilayer overlay modulation included within the received signals.

Orbital angular momentum in combination with optical polarization isexploited within the circuit of FIG. 97 in order to encode informationin rotation invariant photonic states, so as to guarantee fullindependence of the communication from the local reference frames of thetransmitting unit 9702 and the receiving unit 9704. There are variousways to implement quantum key distribution (QKD), a protocol thatexploits the features of quantum mechanics to guarantee unconditionalsecurity in cryptographic communications with error rate performancesthat are fully compatible with real world application environments.

Encrypted communication requires the exchange of keys in a protectedmanner. This key exchanged is often done through a trusted authority.Quantum key distribution is an alternative solution to the keyestablishment problem. In contrast to, for example, public keycryptography, quantum key distribution has been proven to beunconditionally secure, i.e., secure against any attack, even in thefuture, irrespective of the computing power or in any other resourcesthat may be used. Quantum key distribution security relies on the lawsof quantum mechanics, and more specifically on the fact that it isimpossible to gain information about non-orthogonal quantum stateswithout perturbing these states. This property can be used to establishrandom keys between a transmitter and receiver, and guarantee that thekey is perfectly secret from any third party eavesdropping on the line.

In parallel to the “full quantum proofs” mentioned above, the securityof QKD systems has been put on stable information theoretic footing,thanks to the work on secret key agreements done in the framework ofinformation theoretic cryptography and to its extensions, triggered bythe new possibilities offered by quantum information. Referring now toFIG. 98, within a basic QKD system, a QKD link 9802 is a point to pointconnection between a transmitter 9804 and a receiver 9806 that want toshare secret keys. The QKD link 9802 is constituted by the combinationof a quantum channel 9808 and a classic channel 9810. The transmitter9804 generates a random stream of classical bits and encodes them into asequence of non-orthogonal states of light that are transmitted over thequantum channel 9808. Upon reception of these quantum states, thereceiver 9806 performs some appropriate measurements leading thereceiver to share some classical data over the classical link 9810correlated with the transmitter bit stream. The classical channel 9810is used to test these correlations.

If the correlations are high enough, this statistically implies that nosignificant eavesdropping has occurred on the quantum channel 9808 andthus, that has a very high probability, a perfectly secure, symmetrickey can be distilled from the correlated data shared by the transmitter9804 and the receiver 9806. In the opposite case, the key generationprocess has to be aborted and started again. The quantum keydistribution is a symmetric key distribution technique. Quantum keydistribution requires, for authentication purposes, that the transmitter9804 and receiver 9806 share in advance a short key whose length scalesonly logarithmically in the length of the secret key generated by an OKDsession.

Quantum key distribution on a regional scale has already beendemonstrated in a number of countries. However, free-space optical linksare required for long distance communication among areas which are notsuitable for fiber installation or for moving terminals, including theimportant case of satellite based links. The present approach exploitsspatial transverse modes of the optical beam, in particular of the OAMdegree of freedom, in order to acquire a significant technical advantagethat is the insensitivity of the communication to relevant alignment ofthe user's reference frames. This advantage may be very relevant forquantum key distribution implementation to be upgraded from the regionalscale to a national or continental one, or for links crossing hostileground, and even for envisioning a quantum key distribution on a globalscale by exploiting orbiting terminals on a network of satellites.

The OAM Eigen modes are characterized by a twisted wavefront composed of“

” intertwined helices, where “

” is an integer, and by photons carrying “±

” of (orbital) angular momentum, in addition to the more usual spinangular momentum (SAM) associated with polarization. The potentiallyunlimited value of “

” opens the possibility to exploit OAM also for increasing the capacityof communication systems (although at the expense of increasing also thechannel cross-section size), and terabit classical data transmissionbased on OAM multiplexing can be demonstrated both in free-space andoptical fibers. Such a feature can also be exploited in the quantumdomain, for example to expand the number of qubits per photon, or toachieve new functions, such as the rotational invariance of the qubits.

In a free-space QKD, two users (Alice and Bob) must establish a sharedreference frame (SRF) in order to communicate with good fidelity. Indeedthe lack of a SRF is equivalent to an unknown relative rotation whichintroduces noise into the quantum channel, disrupting the communication.When the information is encoded in photon polarization, such a referenceframe can be defined by the orientations of Alice's and Bob's“horizontal” linear polarization directions. The alignment of thesedirections needs extra resources and can impose serious obstacles inlong distance free space QKD and/or when the misalignment varies intime. As indicated, we can solve this by using rotation invariantstates, which remove altogether the need for establishing a SRF. Suchstates are obtained as a particular combination of OAM and polarizationmodes (hybrid states), for which the transformation induced by themisalignment on polarization is exactly balanced by the effect of thesame misalignment on spatial modes. These states exhibit a globalsymmetry under rotations of the beam around its axis and can bevisualized as space-variant polarization states, generalizing thewell-known azimuthal and radial vector beams, and forming atwo-dimensional Hilbert space. Moreover, this rotation-invariant hybridspace can be also regarded as a decoherence-free subspace of thefour-dimensional OAM-polarization product Hilbert space, insensitive tothe noise associated with random rotations.

The hybrid states can be generated by a particular space-variantbirefringent plate having topological charge at its center, named“q-plate”. In particular, a polarized Gaussian beam (having zero OAM)passing through a q-plate with q=½ will undergo the followingtransformation:

(α|R

+β|R))_(π) ⊗|O)_(O) →α|L)_(π) ⊗|r)_(O) +β|R)_(π) ⊗|l)_(O)

|L>_(π) _(_) and |R>_(π) denote the left and right circular polarizationstates (eigenstates of SAM with eigenvalues “±

”), |0>_(O) represents the transverse Gaussian mode with zero OAM andthe |L>_(O) _(_) and |R>_(O) eigenstates of OAM with |

=1 and with eigenvalues “±

”). The states appearing on the right hand side of equation arerotation-invariant states. The reverse operation to this can be realizedby a second q-plate with the same q. In practice, the q-plate operatesas an interface between the polarization space and the hybrid one,converting qubits from one space to the other and vice versa in auniversal (qubit invariant) way. This in turn means that the initialencoding and final decoding of information in our QKD implementationprotocol can be conveniently performed in the polarization space, whilethe transmission is done in the rotation-invariant hybrid space.

OAM is a conserved quantity for light propagation in vacuum, which isobviously important for communication applications. However, OAM is alsohighly sensitive to atmospheric turbulence, a feature which limits itspotential usefulness in many practical cases unless new techniques aredeveloped to deal with such issues.

Quantum cryptography describes the use of quantum mechanical effects (inparticular quantum communication and quantum computation) to performcryptographic tasks or to break cryptographic systems. Well-knownexamples of quantum cryptography are the use of quantum communication toexchange a key securely (quantum key distribution) and the hypotheticaluse of quantum computers that would allow the breaking of variouspopular public-key encryption and signature schemes (e.g., RSA).

The advantage of quantum cryptography lies in the fact that it allowsthe completion of various cryptographic tasks that are proven to beimpossible using only classical (i.e. non-quantum) communication. Forexample, quantum mechanics guarantees that measuring quantum datadisturbs that data; this can be used to detect eavesdropping in quantumkey distribution.

Quantum key distribution (QKD) uses quantum mechanics to guaranteesecure communication. It enables two parties to produce a shared randomsecret key known only to them, which can then be used to encrypt anddecrypt messages.

An important and unique property of quantum distribution is the abilityof the two communicating users to detect the presence of any third partytrying to gain knowledge of the key. This results from a fundamentalaspect of quantum mechanics: the process of measuring a quantum systemin general disturbs the system. A third party trying to eavesdrop on thekey must in some way measure it, thus introducing detectable anomalies.By using quantum superposition or quantum entanglement and transmittinginformation in quantum states, a communication system can be implementedwhich detects eavesdropping. If the level of eavesdropping is below acertain threshold, a key can be produced that is guaranteed to be secure(i.e. the eavesdropper has no information about it), otherwise no securekey is possible and communication is aborted.

The security of quantum key distribution relies on the foundations ofquantum mechanics, in contrast to traditional key distribution protocolwhich relies on the computational difficulty of certain mathematicalfunctions, and cannot provide any indication of eavesdropping orguarantee of key security.

Quantum key distribution is only used to reduce and distribute a key,not to transmit any message data. This key can then be used with anychosen encryption algorithm to encrypt (and decrypt) a message, which istransmitted over a standard communications channel. The algorithm mostcommonly associated with QKD is the one-time pad, as it is provablysecure when used with a secret, random key.

Quantum communication involves encoding information in quantum states,or qubits, as opposed to classical communication's use of bits. Usually,photons are used for these quantum states and thus is applicable withinoptical communication systems. Quantum key distribution exploits certainproperties of these quantum states to ensure its security. There areseveral approaches to quantum key distribution, but they can be dividedinto two main categories, depending on which property they exploit. Thefirst of these are prepare and measure protocol. In contrast toclassical physics, the act of measurement is an integral part of quantummechanics. In general, measuring an unknown quantum state changes thatstate in some way. This is known as quantum indeterminacy, and underliesresults such as the Heisenberg uncertainty principle, informationdistribution theorem, and no cloning theorem. This can be exploited inorder to detect any eavesdropping on communication (which necessarilyinvolves measurement) and, more importantly, to calculate the amount ofinformation that has been intercepted. Thus, by detecting the changewithin the signal, the amount of eavesdropping or information that hasbeen intercepted may be determined by the receiving party.

The second category involves the use of entanglement based protocols.The quantum states of two or more separate objects can become linkedtogether in such a way that they must be described by a combined quantumstate, not as individual objects. This is known as entanglement, andmeans that, for example, performing a measurement on one object affectsthe other object. If an entanglement pair of objects is shared betweentwo parties, anyone intercepting either object alters the overallsystem, revealing the presence of a third party (and the amount ofinformation that they have gained). Thus, again, undesired reception ofinformation may be determined by change in the entangled pair of objectsthat is shared between the parties when intercepted by an unauthorizedthird party.

One example of a quantum key distribution (QKD) protocol is the BB84protocol. The BB84 protocol was originally described using photonpolarization states to transmit information. However, any two pairs ofconjugate states can be used for the protocol, and optical fiber-basedimplementations described as BB84 can use phase-encoded states. Thetransmitter (traditionally referred to as Alice) and the receiver(traditionally referred to as Bob) are connected by a quantumcommunication channel which allows quantum states to be transmitted. Inthe case of photons, this channel is generally either an optical fiber,or simply free-space, as described previously with respect to FIG. 97.In addition, the transmitter and receiver communicate via a publicclassical channel, for example using broadcast radio or the Internet.Neither of these channels needs to be secure. The protocol is designedwith the assumption that an eavesdropper (referred to as Eve) caninterfere in any way with both the transmitter and receiver.

Referring now to FIG. 99, the security of the protocol comes fromencoding the information in non-orthogonal states. Quantum indeterminacymeans that these states cannot generally be measured without disturbingthe original state. BB84 uses two pair of states 9902, each pairconjugate to the other pair to form a conjugate pair 9904. The twostates 9902 within a pair 9904 are orthogonal to each other. Pairs oforthogonal states are referred to as a basis. The usual polarizationstate pairs used are either the rectilinear basis of vertical (0degrees) and horizontal (90 degrees), the diagonal basis of 45 degreesand 135 degrees, or the circular basis of left handedness and/or righthandedness. Any two of these basis are conjugate to each other, and soany two can be used in the protocol. In the example of FIG. 100,rectilinear basis are used at 10002 and 10004, respectively, anddiagonal basis are used at 10006 and 10008.

The first step in BB84 protocol is quantum transmission. Referring nowto FIG. 101 wherein there is illustrated a flow diagram describing theprocess, wherein the transmitter creates a random bit (0 or 1) at step10102, and randomly selects at 10104 one of the two basis, eitherrectilinear or diagonal, to transmit the random bit. The transmitterprepares at step 10106 a photon polarization state depending both on thebit value and the selected basis, as shown in FIG. 55. So, for example,a 0 is encoded in the rectilinear basis (+) as a vertical polarizationstate and a 1 is encoded in a diagonal basis (X) as a 135 degree state.The transmitter transmits at step 10108 a single proton in the statespecified to the receiver using the quantum channel. This process isrepeated from the random bit stage at step 10102 with the transmitterrecording the state, basis, and time of each photon that is sent overthe optical link.

According to quantum mechanics, no possible measurement distinguishesbetween the four different polarization states 10002 through 10008 ofFIG. 100, as they are not all orthogonal. The only possible measurementis between any two orthogonal states (and orthonormal basis). So, forexample, measuring in the rectilinear basis gives a result of horizontalor vertical. If the photo was created as horizontal or vertical (as arectilinear eigenstate), then this measures the correct state, but if itwas created as 45 degrees or 135 degrees (diagonal eigenstate), therectilinear measurement instead returns either horizontal or vertical atrandom. Furthermore, after this measurement, the proton is polarized inthe state it was measured in (horizontal or vertical), with all of theinformation about its initial polarization lost.

Referring now to FIG. 102, as the receiver does not know the basis thephotons were encoded in, the receiver can only select a basis at randomto measure in, either rectilinear or diagonal. At step 10202, thetransmitter does this for each received photon, recording the timemeasurement basis used and measurement result at step 10204. Δt step10206, a determination is made if there are further protons present and,if so, control passes back to step 10202. Once inquiry step 10206determines the receiver had measured all of the protons, the transceivercommunicates at step 10208 with the transmitter over the publiccommunications channel. The transmitter broadcast the basis for eachphoton that was sent at step 10210 and the receiver broadcasts the basiseach photon was measured in at step 10212. Each of the transmitter andreceiver discard photon measurements where the receiver used a differentbasis at step 10214 which, on average, is one-half, leaving half of thebits as a shared key, at step 10216. This process is more fullyillustrated in FIG. 103.

The transmitter transmits the random bit 01101001. For each of thesebits respectively, the transmitter selects the sending basis ofrectilinear, rectilinear, diagonal, rectilinear, diagonal, diagonal,diagonal, and rectilinear. Thus, based upon the associated random bitsselected and the random sending basis associated with the signal, thepolarization indicated in line 10202 is provided. Upon receiving thephoton, the receiver selects the random measuring basis as indicated inline 10304. The photon polarization measurements from these basis willthen be as indicated in line 10306. A public discussion of thetransmitted basis and the measurement basis are discussed at 10308 andthe secret key is determined to be 0101 at 10310 based upon the matchingbases for transmitted photons 1, 3, 6, and 8.

Referring now to FIG. 104, there is illustrated the process fordetermining whether to keep or abort the determined key based uponerrors detected within the determined bit string. To check for thepresence of eavesdropping, the transmitter and receiver compare acertain subset of their remaining bit strings at step 10402. If a thirdparty has gained any information about the photon's polarization, thisintroduces errors within the receiver's measurements. If more than Pbits differ at inquiry step 10404, the key is aborted at step 10406, andthe transmitter and receiver try again, possibly with a differentquantum channel, as the security of the key cannot be guaranteed. P ischosen so that if the number of bits that is known to the eavesdropperis less than this, privacy amplification can be used to reduce theeavesdropper's knowledge of the key to an arbitrarily small amount byreducing the length of the key. If inquiry step 10404 determines thatthe number of bits is not greater than P, then the key may be used atstep 10408.

The E91 protocol comprises another quantum key distribution scheme thatuses entangled pairs of protons. This protocol may also be used withentangled pairs of protons using orbital angular momentum processing,Laguerre Gaussian processing, Hermite Gaussian processing or processingusing any orthogonal functions for Q-bits. The entangled pairs can becreated by the transmitter, by the receiver, or by some other sourceseparate from both of the transmitter and receiver, including aneavesdropper. The photons are distributed so that the transmitter andreceiver each end up with one photon from each pair. The scheme relieson two properties of entanglement. First, the entangled states areperfectly correlated in the sense that if the transmitter and receiverboth measure whether their particles have vertical or horizontalpolarizations, they always get the same answer with 100 percentprobability. The same is true if they both measure any other pair ofcomplementary (orthogonal) polarizations. However, the particularresults are not completely random. It is impossible for the transmitterto predict if the transmitter, and thus the receiver, will get verticalpolarizations or horizontal polarizations. Second, any attempt ateavesdropping by a third party destroys these correlations in a way thatthe transmitter and receiver can detect. The original Ekert protocol(E91) consists of three possible states and testing Bell inequalityviolation for detecting eavesdropping.

Presently, the highest bit rate systems currently using quantum keydistribution demonstrate the exchange of secure keys at 1 Megabit persecond over a 20 kilometer optical fiber and 10 Kilobits per second overa 100 kilometer fiber.

The longest distance over which quantum key distribution has beendemonstrated using optical fiber is 148 kilometers. The distance is longenough for almost all of the spans found in today's fiber-opticnetworks. The distance record for free-space quantum key distribution is144 kilometers using BB84 enhanced with decoy states.

Referring now to FIG. 105, there is illustrated a functional blockdiagram of a transmitter 10502 and receiver 10504 that can implementalignment of free-space quantum key distribution. The system canimplement the BB84 protocol with decoy states. The controller 10506enables the bits to be encoded in two mutually unbiased bases Z={|0>,|1>} and X={|+>, |−}, where |0> and |1> are two orthogonal statesspanning the qubit space and |±

=1/√2 (|0

±1

). The transmitter controller 10506 randomly chooses between the Z and Xbasis to send the classical bits 0 and 1. Within hybrid encoding, the Zbasis corresponds to {|L

_(π)⊗|r

_(O), |R

_(π)⊗|l

_(O)} while the X basis states correspond to 1/√2 (|L

_(π)⊗|r

_(O)±|R

_(π)⊗|l

_(O)). The transmitter 10502 uses four different polarized attenuatedlasers 10508 to generate quantum bits through the quantum bit generator10510. Photons from the quantum bit generator 41050 are delivered via asingle mode fiber 10512 to a telescope 10514. Polarization states |H>,|V>, |R>, |L> are transformed into rotation invariant hybrid states bymeans of a q-plate 10516 with q=½. The photons can then be transmittedto the receiving station 10504 where a second q-plate transform 10518transforms the signals back into the original polarization states |H>,|V>, |R>, |L>, as defined by the receiver reference frame. Qubits canthen be analyzed by polarizers 10520 and single photon detectors 10522.The information from the polarizers 10520 and photo detectors 10522 maythen be provided to the receiver controller 10524 such that the shiftedkeys can be obtained by keeping only the bits corresponding to the samebasis on the transmitter and receiver side as determined bycommunications over a classic channel between the transceivers 10526,10528 in the transmitter 10502 and receiver 10504.

Referring now to FIG. 106, there is illustrated a network cloud basedquantum key distribution system including a central server 10602 andvarious attached nodes 10604 in a hub and spoke configuration. Trends innetworking are presenting new security concerns that are challenging tomeet with conventional cryptography, owing to constrained computationalresources or the difficulty of providing suitable key management. Inprinciple, quantum cryptography, with its forward security andlightweight computational footprint, could meet these challenges,provided it could evolve from the current point to point architecture toa form compatible with multimode network architecture. Trusted quantumkey distribution networks based on a mesh of point to point links lacksscalability, require dedicated optical fibers, are expensive and notamenable to mass production since they only provide one of thecryptographic functions, namely key distribution needed for securecommunications. Thus, they have limited practical interest.

A new, scalable approach such as that illustrated in FIG. 106 providesquantum information assurance that is network based quantumcommunications which can solve new network security challenges. In thisapproach, a BB84 type quantum communication between each of N clientnodes 10604 and a central sever 10602 at the physical layer support aquantum key management layer, which in turn enables secure communicationfunctions (confidentiality, authentication, and nonrepudiation) at theapplication layer between approximately N2 client pairs. This networkbased communication “hub and spoke” topology can be implemented in anetwork setting, and permits a hierarchical trust architecture thatallows the server 10602 to act as a trusted authority in cryptographicprotocols for quantum authenticated key establishment. This avoids thepoor scaling of previous approaches that required a pre-existing trustrelationship between every pair of nodes. By making a server 10602, asingle multiplex QC (quantum communications) receiver and the clientnodes 10604 QC transmitters, this network can simplify complexity acrossmultiple network nodes. In this way, the network based quantum keydistribution architecture is scalable in terms of both quantum physicalresources and trust. One can at time multiplex the server 10602 withthree transmitters 10604 over a single mode fiber, larger number ofclients could be accommodated with a combination of temporal andwavelength multiplexing as well as orbital angular momentum multiplexedwith wave division multiplexing to support much higher clients.

Referring now to FIGS. 107 and 108, there are illustrated variouscomponents of multi-user orbital angular momentum based quantum keydistribution multi-access network. FIG. 107 illustrates a high speedsingle photon detector 10702 positioned at a network node that can beshared between multiple users 10704 using conventional networkarchitectures, thereby significantly reducing the hardware requirementsfor each user added to the network. In an embodiment, the single photondetector 10702 may share up to 64 users. This shared receiverarchitecture removes one of the main obstacles restricting thewidespread application of quantum key distribution. The embodimentpresents a viable method for realizing multi-user quantum keydistribution networks with resource efficiency.

Referring now also to FIG. 108, in a nodal quantum key distributionnetwork, multiple trusted repeaters 10802 are connected via point topoint links 10804 between node 10806. The repeaters are connected viapoint to point links between a quantum transmitter and a quantumreceiver. These point to point links 10804 can be realized using longdistance optical fiber lengths and may even utilize ground to satellitequantum key distribution communication. While point to point connections10804 are suitable to form a backbone quantum core network, they areless suitable to provide the last-mile service needed to give amultitude of users access to the quantum key distributioninfrastructure. Reconfigurable optical networks based on opticalswitches or wavelength division multiplexing may achieve more flexiblenetwork structures, however, they also require the installation of afull quantum key distribution system per user which is prohibitivelyexpensive for many applications.

The quantum key signals used in quantum key distribution need onlytravel in one direction along a fiber to establish a secure key betweenthe transmitter and the receiver. Single photon quantum key distributionwith the sender positioned at the network node 10806 and the receiver atthe user premises therefore lends itself to a passive multi-user networkapproach. However, this downstream implementation has two majorshortcomings. Firstly, every user in the network requires a singlephoton detector, which is often expensive and difficult to operate.Additionally, it is not possible to deterministically address a user.All detectors, therefore, have to operate at the same speed as atransmitter in order not to miss photons, which means that most of thedetector bandwidth is unused.

Most systems associated with a downstream implementation can beovercome. The most valuable resource should be shared by all users andshould operate at full capacity. One can build an upstream quantumaccess network in which the transmitters are placed at the end userlocation and a common receiver is placed at the network node. This way,an operation with up to 64 users is feasible, which can be done withmulti-user quantum key distribution over a 1×64 passive opticalsplitter.

The above described QKD scheme is applicable to twisted pair, coaxialcable, fiber optic, RF satellite, RF broadcast, RF point-to point, RFpoint-to-multipoint, RF point-to-point (backhaul), RF point-to-point(fronthaul to provide higher throughput CPRI interface forcloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE. The techniques would be usefulfor combating denial of service attacks by routing communications viaalternate links in case of disruption, as a technique to combat TrojanHorse attacks which does not require physical access to the endpointsand as a technique to combat faked-state attacks, phase remappingattacks and time-shift attacks.

Thus, using various configurations of the above described orbitalangular momentum processing, multi-layer overlay modulation, and quantumkey distribution within various types of communication networks and moreparticularly optical fiber networks and free-space optic communicationnetwork, a variety of benefits and improvements in system bandwidth andcapacity may be achieved.

OAM Based Networking Functions

In addition to the potential applications for static point to point datatransmission, the unique way front structure of OAM beams may alsoenable some networking functions by manipulating the phase usingreconfigurable spatial light modulators (SLMs) or other light projectingtechnologies.

Data Swapping

Data exchange is a useful function in an OAM-based communication system.A pair of data channels on different OAM states can exchange their datain a simple manner with the assistance of a reflective phase hologram asillustrated in FIG. 109. If two OAM beams 10902, 10904, e.g., OAM beamswith

=+L₁ and +L₂, which carry two independent data streams 10906, 10908, arelaunched onto a reflective SLM 10910 loaded with a spiral phase patternwith an order of −(L1+L2), the data streams will swap between the twoOAM channels. The phase profile of the SLM will change these two OAMbeams to

=−

₂ and

=−

₁, respectively. In addition, each OAM beam will change to its oppositecharge under the reflection effect. As a result, the channel on

=+

₁ is switched to

=+

₂ and vice versa, which indicates that the data on the two OAM channelsis exchanged. FIG. 109 shows the data exchange between

=+6 10912 and

=8 10914 using a phase pattern on the order of

=−14 on a reflective SLM 10910. A power penalty of approximately 0.9 dBis observed when demonstrating this in the experiment.

An experiment further demonstrated that the selected data swappingfunction can handle more than two channels. Among multiple multiplexedOAM beams, any two OAM beams can be selected to swap their data withoutaffecting the other channels. In general, reconfigurable opticaladd/drop multiplexers (ROADM) are important function blocks in WDMnetworks. A WDM RODAM is able to selectively drop a given wavelengthchannel and add in a different channel at the same wavelength withouthaving to detect all pass-through channels. A similar scheme can beimplemented in an OAM multiplexed system to selectively drop and add adata channel carried on a given OAM beam. One approach to achieve thisfunction is based on the fact that OAM beams generally have a distinctintensity profile when compared to a fundamental Gaussian beam.

Referring now to FIGS. 110 and 111 there is illustrated the manner forusing a ROADM for exchanging data channels. The example of FIG. 111illustrates SLM's and spatial filters. The principle of an OAM-basedROADM uses three stages: down conversion, add/drop and up conversion.The down conversion stage transforms at step 11002 the input multiplexedOAM modes 11102 (donut like transverse intensity profiles 11104) into aGaussian light beam with

=0 (a spotlight transverse intensity profile 11106). After the downconversion at step 11002, the selected OAM beam becomes a Gaussian beamwhile the other beams remain OAM but have a different

state. The down converted beams 11106 are reflected at step 11004 by aspecially designed phase pattern 11108 that has different gratings inthe center and in the outer ring region. The central and outer regionsare used to redirect the Gaussian beam 11106 in the center (containingthe drop channel 11110) and the OAM beams with a ring-shaped (containingthe pass-through channels) in different directions. Meanwhile, anotherGaussian beam 11112 carrying a new data stream can be added to thepass-through OAM beams (i.e., add channel). Following the selectivemanipulation, an up conversion process is used at step 11006 fortransforming the Gaussian beam back to an OAM beam. This processrecovers the

states of all of the beams. FIG. 92 illustrates the images of each stepin the add/drop of a channel carried by an OAM beam with

=+2. Some other networking functions in OAM based systems have also beendemonstrated including multicasting, 2 by 2 switching, polarizationswitching and mode filtering.

In its fundamental form, a beam carrying OAM has a helical phase frontthat creates orthogonality and hence is distinguishable from other OAMstates. Although other mode groups (e.g., Hermite-Gaussian modes, etc.)also have orthogonality and can be used for mode multiplexing, OAM hasthe convenient advantage of its circular symmetry which is matched tothe geometry of most optical systems. Indeed, many free-space data linkdemonstrations attempt to use OAM-carrying modes since such modes havecircular symmetry and tend to be compatible with commercially availableoptical components. Therefore, one can consider that OAM is used more asa technical convenience for efficient multiplexing than as a necessarily“better” type of modal set.

The use of OAM multiplexing in fiber is potentially attractive. In aregular few mode fiber, hybrid polarized OAM modes can be considered asfiber eigenmodes. Therefore, OAM modes normally have less temporalspreading as compared to LP mode basis, which comprise two eigenmodecomponents each with a different propagation constant. As for thespecially designed novel fiber that can stably propagate multiple OAMstates, potential benefits could include lower receiver complexity sincethe MIMO DSP is not required. Progress can be found in developingvarious types of fiber that are suitable for OAM mode transmission.Recently demonstrated novel fibers can support up to 16 OAM states.Although they are still in the early stages, there is the possibilitythat further improvement of performance (i.e., larger number of“maintained” modes and lower power loss) will be achieved.

OAM multiplexing can be useful for communications in RF communicationsin a different way than the traditional spatial multiplexing. For atraditional spatial multiplexing system, multiple spatially separatedtransmitter and receiver aperture pairs are adopted for the transmissionof multiple data streams. As each of the antenna elements receives adifferent superposition of the different transmitted signals, each ofthe original channels can be demultiplexed through the use of electronicdigital signal processing. The distinction of each channel relies on thespatial position of each antenna pair. However, OAM multiplexing isimplemented such that the multiplexed beams are completely coaxialthroughout the transmission medium, and only one transmitter andreceiver aperture (although with certain minimum aperture sizes) isused. Due to the OAM beam orthogonality provided by the helical phasefront, efficient demultiplexing can be achieved without the assist offurther digital signal post-processing to cancel channel interference.

Many of the demonstrated communication systems with OAM multiplexing usebulky and expensive components that are not necessarily optimized forOAM operation. As was the case for many previous advances in opticalcommunications, the future of OAM would greatly benefit from advances inthe enabling devices and subsystems (e.g., transmitters,(de)multiplexers and receivers). Particularly with regard tointegration, this represents significant opportunity to reduce cost andsize and to also increase performance.

Orthogonal beams using for example OAM, Hermite Gaussian, LaguerreGaussian, spatial Bessel, Prolate spheroidal or other types oforthogonal functions may be multiplexed together to increase the amountof information transmitted over a single communications link. Thestructure for multiplexing the beams together may use a number ofdifferent components. Examples of these include spatial light modulators(SLMs); micro electromechanical systems (MEMs); digital light processors(DLPs); amplitude masks; phase masks; spiral phase plates; Fresnel zoneplates; spiral zone plates; spiral phase plates and phase plates.

Multiplexing Using Holograms

Referring now to FIG. 113, there is illustrated a configuration ofgeneration circuitry for the generation of an OAM twisted beam using ahologram within a micro-electrical mechanical device. Configurationssuch as this may be used for multiplexing multiple OAM twisted beamstogether. A laser 11302 generates a beam having a wavelength ofapproximately 543 nm. This beam is focused through a telescope 11304 andlens 11306 onto a mirror/system of mirrors 11308. The beam is reflectedfrom the mirrors 11308 into a DMD 11310. The DMD 11310 has programmed into its memory a one or more forked holograms 11312 that generate adesired OAM twisted beam 11313 having any desired information encodedinto the OAM modes of the beam that is detected by a CCD 11314. Theholograms 11312 are loaded into the memory of the DMD 11310 anddisplayed as a static image. In the case of 1024×768 DMD array, theimages must comprise 1024 by 768 images. The control software of the DMD11310 converts the holograms into .bmp files. The holograms may bedisplayed singly or as multiple holograms displayed together in order tomultiplex particular OAM modes onto a single beam. The manner ofgenerating the hologram 11312 within the DMD 11310 may be implemented ina number of fashions that provide qualitative differences between thegenerated OAM beam 11313. Phase and amplitude information may be encodedinto a beam by modulating the position and width of a binary amplitudegrating used as a hologram. By realizing such holograms on a DMD thecreation of HG modes, LG modes, OAM vortex mode or any angular mode maybe realized. Furthermore, by performing switching of the generated modesat a very high speed, information may be encoded within the helicity'sthat are dynamically changing to provide a new type of helicitymodulation. Spatial modes may be generated by loading computer-generatedholograms onto a DMD. These holograms can be created by modulating agrating function with 20 micro mirrors per each period.

Rather than just generating an OAM beam 11313 having only a single OAMvalue included therein, multiple OAM values may be multiplexed into theOAM beam in a variety of manners as described herein below. The use ofmultiple OAM values allows for the incorporation of differentinformation into the light beam. Programmable structured light providedby the DLP allows for the projection of custom and adaptable patterns.These patterns may be programmed into the memory of the DLP and used forimparting different information through the light beam. Furthermore, ifthese patterns are clocked dynamically a modulation scheme may becreated where the information is encoded in the helicities of thestructured beams.

Referring now to FIG. 114, rather than just having the laser beam 11402shine on a single hologram multiple holograms 11404 may be generated bythe DMD 4410. FIG. 114 illustrates an implementation wherein a 4×3 arrayof holograms 11404 are generated by the DMD 4410. The holograms 11404are square and each edge of a hologram lines up with an edge of anadjacent hologram to create the 4×3 array. The OAM values provided byeach of the holograms 11404 are multiplexed together by shining the beam11402 onto the array of holograms 11404. Several configurations of theholograms 11404 may be used in order to provide differing qualities ofthe OAM beam 11313 and associated modes generated by passing a lightbeam through the array of holograms 11404.

Referring now to FIG. 115 there is illustrated an alternative way ofmultiplexing various OAM modes together. An X by Y array of holograms11502 has each of the hologram 11502 placed upon a black (dark)background 11504 in order to segregate the various modes from eachother. In another configuration illustrated in FIG. 116, the holograms11602 are placed in a hexagonal configuration with the background in theoff (black) state in order to better segregate the modes.

FIG. 117 illustrates yet another technique for multiplexing multiple OAMmodes together wherein the holograms 11702 are cycled through in a loopsequence by the DMD 11310. In this example modes T₀-T₁₁ are cycledthrough and the process repeats by returning back to mode T₀. Thisprocess repeats in a continuous loop in order to provide an OAM twistedbeam with each of the modes multiplex therein.

In addition to providing integer OAM modes using holograms within theDMD, fractional OAM modes may also be presented by the DMD usingfractional binary forks as illustrated in FIG. 118. FIG. 118 illustratesfractional binary forks for generating fractional OAM modes of 0.25,0.50, 0.75, 1.25, 1.50 and 1.75 with a light beam.

Referring now to FIG. 119-132, there are illustrated the resultsachieved from various configurations of holograms program within thememory of a DMD. FIG. 119 illustrates the configuration at 11902 havingno hologram separation on a white background producing the OAM modeimage 11904. FIG. 120 uses a configuration 12002 consisting of circularholograms 11904 having separation on a white background. The OAM modeimage 11906 that is provided therefrom is also illustrated. Bright modeseparation yields less light and better mode separation.

FIG. 121 illustrates a configuration 12102 having square holograms withno separation on a black background. The configuration 12102 generatesthe OAM mode image 12104. FIG. 122 illustrates the configuration ofcircular holograms (radius ˜256 pixels) that are separated on a blackbackground. This yields the OAM mode image 12204. Dark mode separationyields more light in the OAM image 12204 and has slightly better modeseparation.

FIG. 123 illustrates a configuration 12302 having a bright backgroundand circular hologram (radius ˜256 pixels) separation yielding an OAMmode image 12304. FIG. 124 illustrates a configuration 12402 usingcircular holograms (radius ˜256 pixels) having separation on a blackbackground to yield the OAM mode image 12404. The dark mode separationyields more light and has a slightly worse mode separation within theOAM mode images.

FIG. 125 illustrates a configuration 12502 including circular holograms(radius ˜256 pixels) in a hexagonal distribution on a bright backgroundyielding an OAM mode image 12504. FIG. 126 illustrates at 12602 smallcircular holograms (radius ˜256 pixels) in a hexagonal distribution on abright background that yields and OAM mode image 12604. The largerholograms with brighter backgrounds yield better OAM mode separationimages.

Referring now to FIG. 127, there is illustrated a configuration 12702 ofcircular holograms (radius ˜256 pixels) in a hexagonal distribution on adark background with each of the holograms having a radius ofapproximately 256 pixels. This configuration 12702 yields the OAM modeimage 12704. FIG. 128 illustrates the use of small holograms (radius˜256 pixels) having a radius of approximately 190 pixels arranged in ahexagonal distribution on a black background that yields the OAM modeimage 12804. Larger holograms (radius of approximately 256 pixels)having a dark background yields worse OAM mode separation within the OAMmode images.

FIG. 129 illustrates a configuration 12902 of small holograms (radius ofapproximately 190 pixels) in a hexagonal separated distribution on adark background that yields the OAM mode image 12904. FIG. 130illustrates a configuration 13002 of small holograms (radius ˜256pixels) in a hexagonal distribution that are close together on a darkbackground that yields the OAM mode image 13004. The larger darkboundaries (FIG. 129) yield worse OAM mode image separation than asmaller dark boundary.

FIG. 131 illustrates a configuration 13102 of small holograms (radius˜256 pixels) in a separated hexagonal configuration on a brightbackground yielding OAM mode image 13104. FIG. 63 illustrates aconfiguration 6302 of small holograms (radius ˜256 pixels) more closelyspaced in a hexagonal configuration on a bright background yielding OAMmode image 6304. The larger bright boundaries (FIG. 131) yield a betterOAM mode separation.

Additional illustrations of holograms, namely reduced binary hologramsare illustrated in FIGS. 133-136. FIG. 133 illustrates reduced binaryholograms having a radius equal to 100 micro mirrors and a period of 50for various OAM modes. Similarly, OAM modes are illustrated for reducedbinary for holograms having a radius of 50 micro mirrors and a period of50 (FIG. 134); a radius of 100 micro mirrors and a period of 100 (FIG.135) and a radius of 50 micro mirrors and a period of 50 (FIG. 136).

The illustrated data with respect to the holograms of FIGS. 119-136demonstrates that full forked gratings yield a great deal of scatteredlight. Finer forked gratings yield better define modes within OAMimages. By removing unnecessary light from the hologram (white regions)there is a reduction in scatter. Holograms that are larger and havefewer features (more dark zones) having a hologram diameter of 200 micromirrors provide overlapping modes and strong intensity. Similarconfigurations using 100 micro mirrors also demonstrate overlappingmodes and strong intensity. Smaller holograms having smaller radiibetween 100-200 micro mirrors and periods between 50 and 100 generatedby a DLP produce better defined modes and have stronger intensity thanlarger holograms with larger radii in periods. Smaller holograms havingmore features (dark zones with hologram diameters of 200 micro mirrorsprovide well-defined modes with strong intensity. However, hundred micromirror diameter holograms while providing well-defined modes provideweaker intensity. Thus, good, compact hologram sizes are between 100-200micro mirrors with zone periods of between 50 and 100. Larger hologramshave been shown to provide a richer OAM topology.

Referring now to FIGS. 139 and 140, there are illustrated a blockdiagram of a circuit for generating a muxed and multiplexed data streamcontaining multiple new Eigen channels (FIG. 139) for transmission overa communications link (free space, fiber, RF, etc.), and a flow diagramof the operation of the circuit (FIG. 140). Multiple data streams 13902are received at step 14002 and input to a modulator circuit 13904. Themodulator circuit 13904 modulates a signal with the data stream at step14004 and outputs these signals to the orthogonal function circuit13906. The orthogonal function circuit 13906 applies a differentorthogonal function to each of the data streams at step 14006. Theseorthogonal functions may comprise orbital angular momentum functions,Hermite Gaussian functions, Laguerre Gaussian functions, prolatespheroidal functions, Bessel functions or any other types of orthogonalfunctions. Each of the data streams having an orthogonal functionapplied thereto are applied to the mux circuit 13098. The mux circuit13098 performs a spatial combination of multiple orthogonal signals ontoa same physical bandwidth at step 14008. Thus, a single signal willinclude multiple orthogonal data streams that are all located within thesame physical bandwidth. A plurality of these muxed signals are appliedto the multiplexing circuit 13910. The multiplexing circuit 13910multiplexes multiple muxed signals onto a same frequency or wavelengthat step 14010. Thus, the multiplexing circuit 13910 temporallymultiplexes multiple signals onto the same frequency or wavelength. Themuxed and multiplexed signal is provided to a transmitter 13912 suchthat the signal 13914 may be transmitted at step 14012 over acommunications link (Fiber, FSO, RF, etc.).

Referring now to FIGS. 141 and 142, there is illustrated a block diagram(FIG. 141) of the receiver side circuitry and a flow diagram (FIG. 142)of the operation of the receiver side circuitry associated with thecircuit of FIG. 139. A received signal 14102 is input to the receiver14104 at step 14202. The receiver 14014 provides the received signal14102 to the de-multiplexer circuit 14106. The de-multiplexer circuit14106 separates the temporally multiplexed received signal 14102 intomultiple muxed signals at step 14204 and provides them to the de-muxcircuit 14108. As discussed previously with respect to FIGS. 139 and140, the de-multiplexer circuit 14106 separates the muxed signals thatare temporally multiplexed onto a same frequency or wavelength. Thede-mux circuit 14108 separates (de-muxes) the multiple orthogonal datastreams at step 14206 from the same physical bandwidth. The multipleorthogonal data streams are provided to the orthogonal function circuit14110 that removes the orthogonal function at step 14208. The individualdata streams may then be demodulated within the demodulator circuit14112 at step 14210 and the multiple data streams 14114 provided foruse.

MIMO+MDM

A further manner for increasing data transmission bandwidth is thecombination of multiple-input multiple-output (MIMO)-based spatialmultiplexing and Mode Division Multiplexing (MDM) using Hermite-Gaussian(HG), Laguerre-Gaussian (LG) or other orthogonal function processedbeams. The LG beams carry orbital angular momentum (OAM) and thereforesuch multiplexing technique can also be called OAM multiplexing. Such acombined MIMO+MDM multiplexing can enhance the performance of free-spacePoint-to-Point communications systems by fully exploiting the advantagesof each multiplexing technique. This can be done at both RF as well asoptical frequencies. Inter-channel crosstalk effects can be minimized bythe OAM beams' inherent orthogonality and by the use of MIMO signalprocessing. OAM and MIMO-based spatial multiplexing can be compatiblewith and complement each other, thereby providing potential for a denseor super massive compactified MIMO system.

When multiple input/multiple output (MIMO) systems were described in themid-to-late 1990s by Dr. G. Foschini and Dr. A. Paulraj, the astonishingbandwidth efficiency of such techniques seemed to be in violation of theShannon limit. But, there is no violation of the Shannon limit becausethe diversity and signal processing employed with MIMO transforms apoint-to-point single channel into multiple parallel or matrix channels,hence in effect multiplying the capacity. MIMO offers higher data ratesas well as spectral efficiency. This is more particularly So illustratedin FIG. 143 wherein a single transmitting antenna 14302 transmits to asingle receiving antenna 14304 using a total power signal P_(total). TheMIMO system illustrated in FIG. 144 provides the same total power signalP_(total) to a multi-input transmitter consisting of a plurality ofantennas 14402. The receiver includes a plurality of antennas 14404 forreceiving the transmitted signal. Many standards have alreadyincorporated MIMO. ITU uses MIMO in the High Speed Downlink PacketAccess (HSPDA), part of the UMTS standard. MIMO is also part of the802.11n standard used by wireless routers as well as 802.16 for MobileWiMax, LTE, LTE Advanced and future 5G standards.

A traditional communications link, which is called asingle-in-single-out (SISO) channel as shown in FIG. 143, has onetransmitter 14302 and one receiver 14304. But instead of a singletransmitter and a single receiver several transmitters 14402 andreceivers 14404 may be used as shown in FIG. 144. The SISO channel thusbecomes a multiple-in-multiple-out, or a MIMO channel; i.e. a channelthat has multiple transmitters and multiple receivers.

What does MIMO offer over a traditional SISO channel? To examine thisquestion, we will first look at the capacity of a SISO link, which isspecified in the number of bits that can be transmitted over it asmeasured by the very important metric, (b/s/Hz).

The capacity of a SISO link is a function simply of the channel SNR asgiven by the Equation: C=log₂(1+SNR). This capacity relationship was ofcourse established by Shannon and is also called theinformation-theoretic capacity. The SNR in this equation is defined asthe total power divided by the noise power. The capacity is increasingas a log function of the SNR, which is a slow increase. Clearlyincreasing the capacity by any significant factor takes an enormousamount of power in a SISO channel. It is possible to increase thecapacity instead by a linear function of power with MIMO.

With MIMO, there is a different paradigm of channel capacity. If sixantennas are added on both transmit and receive side, the same capacitycan be achieved as using 100 times more power than in the SISO case. Thetransmitter and receiver are more complex but have no increase in powerat all. The same performance is achieved in the MIMO system as isachieved by increasing the power 100 times in a SISO system.

In FIG. 145, the comparison of SISO and MIMO systems using the samepower. MIMO capacity 14502 increases linearly with the number ofantennas, where SISO/SIMO/MISO systems 14504 all increase onlylogarithmically.

At conceptual level, MIMO enhances the dimensions of communication.However, MIMO is not Multiple Access. It is not like FDMA because all“channels” use the same frequency, and it is not TDMA because allchannels operate simultaneously. There is no way to separate thechannels in MIMO by code, as is done in CDMA and there are no steerablebeams or smart antennas as in SDMA. MIMO exploits an entirely differentdimension.

A MIMO system provides not one channel but multiple channels,N_(R)×N_(T), where N_(T) is the number of antennas on the transmit sideand N_(R), on the receive side. Somewhat like the idea of OFDM, thesignal travels over multiple paths and is recombined in a smart way toobtain these gains.

In FIG. 146 there is illustrated a comparison of a SISO channel 14602with 2 MIMO channels 14604, 14606, (2×2) and (4×4). At SNR of 10 dB, a2×2 MIMO system 14604 offers 5.5 b/s/Hz and whereas a 4×4 MIMO linkoffers over 10 b/s/Hz. This is an amazing increase in capacity withoutany increase in transmit power caused only by increasing the number oftransceivers. Not only that, this superb performance comes in when thereare channel impairments, those that have fading and Doppler.

Extending the single link (SISO) paradigm, it is clear that to increasecapacity, a link can be replicated N times. By using N links, thecapacity is increased by a factor of N. But this scheme also uses Ntimes the power. Since links are often power-limited, the idea of N linkto get N times capacity is not much of a trick. Can the number of linksbe increased but not require extra power? How about if two antennas areused but each gets only half the power? This is what is done in MIMO,more transmit antennas but the total power is not increased. Thequestion is how does this result in increased capacity?

The information-theoretic capacity increase under a MIMO system is quitelarge and easily justifies the increase in complexity as illustrated inFIG. 147. First and second transmitters 14702 transmit to a pair ofreceivers 14704. Each of the transmitters 14702 has a transmission link14706 to an antenna of a receiver 14704. Transmitter TX#1 transmits onlink h11 and h21. Transmitter TX#2 transmits on links h12 and h22. Thisprovides a matrix of transmission capacities according to the matrix:

$H = \begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix}$

And a total transmission capacity of according to the equation:

$C = {\max\limits_{{{tr}{(R_{xx})}} = P_{T}}\mspace{14mu} {\log \mspace{14mu} \det \mspace{14mu} \left\{ {I_{N} + {\frac{I}{\sigma_{n}^{2}}\mspace{14mu} {HR}_{xx}H^{H}}} \right\}}}$

In simple language, MIMO is any link that has multiple transmit andreceive antennas. The transmit antennas are co-located, at a little lessthan half a wavelength apart or more. This figure of the antennaseparation is determined by mutual correlation function of the antennasusing Jakes Model. (See FIG. ______). The receive antennas 14704 arealso part of one unit. Just as in SISO links, the communication isassumed to be between one sender and one receiver. MIMO is also used ina multi-user scenario, similar to the way OFDM can be used for one ormultiple users. The input/output relationship of a SISO channel isdefined as:

r=hs+n

where r is the received signal, s is the sent signal and h, the impulseresponse of the channel is n, the noise. The term h, the impulseresponse of the channel, can be a gain or a loss, it can be phase shiftor it can be time delay, or all of these together. The quantity h can beconsidered an enhancing or distorting agent for the signal SNR.

Referring now to FIG. 148 there are illustrated various types ofmultiple input and multiple output transmission systems. System 14802illustrates a single input single output SISO system. System 14804illustrates a single input multiple output receiver SIMO system. System14806 illustrates a multiple input single output MISO system. Finally, amultiple input multiple output MIMO system is illustrated at 14810. Thechannels of the MIMO system 14810 can be thought of as a matrix channel.

Using the same model a SISO, MIMO channel can now be described as:

R=HS+N

In this formulation, both transmit and receive signals are vectors. Thechannel impulse response h, is now a matrix, H. This channel matrix H iscalled Channel Information. The channel matrix H can be created using apilot signal over a pilot channel in the manner described herein above.The signals on the pilot channel may be sent in a number of differentforms such as HG beams, LG beams or other orthogonal beams of any order.

Dimensionality of Gains in MIMO

The MIMO design of a communications link can be classified in these twoways.

-   -   MIMO using diversity techniques    -   MIMO using spatial-multiplexing techniques        Both of these techniques are used together in MIMO systems. With        first form, Diversity technique, same data is transmitted on        multiple transmit antennas and hence this increases the        diversity of the system.

Diversity means that the same data has traveled through diverse paths toget to the receiver. Diversity increases the reliability ofcommunications. If one path is weak, then a copy of the data received onanother path may be just fine.

FIG. 149 illustrates a source 14902 with data sequence 101 to be sentover a MIMO system with three transmitters. In the diversity form 14904of MIMO, same data, 101 is sent over three different transmitters. Ifeach path is subject to different fading, the likelihood is high thatone of these paths will lead to successful reception. This is what ismeant by diversity or diversity systems. This system has a diversitygain of 3.

The second form uses spatial-multiplexing techniques. In a diversitysystem 14904, the same data is sent over each path. In aspatial-multiplexing system 14906, the data 1,0,1 is multiplexed on thethree channels. Each channel carries different data, similar to the ideaof an OFDM signal. Clearly, by multiplexing the data, the datathroughput or the capacity of the channel is increased, but thediversity gain is lost. The multiplexing has tripled the data rate, sothe multiplexing gain is 3 but diversity gain is now 1. Whereas in adiversity system 14904 the gain comes in form of increased reliability,in a spatial-multiplexing system 14906, the gain comes in the form of anincreased data rate.

Characterizing a MIMO Channel

When a channel uses multiple receive antennas, N_(R), and multipletransmit antennas, N_(T), the system is called a multiple-input,multiple output (MIMO) system.

When N_(T)=N_(R)=1, a SISO system.When N_(T)>1 and N_(R)=1, called a MISO system,When N_(T)=1 and N_(R)>1, called a SIMO system.When N_(T)>1 and N_(R)>1, is a MIMO system.

In a typical SISO channel, the data is transmitted and reception isassumed. As long as the SNR is not changing dramatically, no questionsare asked regarding any information about the channel on a bit by bitbasis. This is referred to as a stable channel. Channel knowledge of aSISO channel is characterized only by its steady-state SNR.

What is meant by channel knowledge for a MIMO channel? Assume a linkwith two transmitters and two receivers on each side. The same symbol istransmitted from each antenna at the same frequency, which is receivedby two receivers. There are four possible paths as shown in FIG. 147.Each path from a transmitter to a receiver has some loss/gain associatedwith it and a channel can be characterized by this loss. A path mayactually be sum of many multipath components but it is characterizedonly by the start and the end points. Since all four channels arecarrying the same symbol, this provides diversity by making up for aweak channel, if any. In FIG. 150 there is illustrated how each channelmay be fading from one moment to the next. At time 32, for example, thefade in channel h₂₁ is much higher than the other three channels.

As the number of antennas and hence the number of paths increase in aMIMO system, there is an associated increase in diversity. Thereforewith t h e increasing numbers of transmitters, all fades can probably becompensated for. With increasing diversity, the fading channel starts tolook like a Gaussian channel, which is a welcome outcome.

The relationship between the received signal in a MIMO system and thetransmitted signal can be represented in a matrix form with the H matrixrepresenting the low-pass channel response h_(ij), which is the channelresponse from the j_(th) antenna to the i_(th) receiver. The matrix H ofsize (N_(R), N_(T)) has N_(R) rows, representing N_(R) received signals,each of which is composed of N_(T) components from N_(T) transmitters.Each column of the H matrix represents the components arriving from onetransmitter to N_(R) receivers.

The H matrix is called the channel information. Each of the matrixentries is a distortion coefficient acting on the transmitted signalamplitude and phase in time-domain. To develop the channel information,a symbol is sent from the first antenna, and a response is noted by allthree receivers. Then the other two antennas do the same thing and a newcolumn is developed by the three new responses.

The H matrix is developed by the receiver. The transmitter typicallydoes not have any idea what the channel looks like and is transmittingblindly. If the receiver then turns around and transmits this matrixback to the transmitter, then the transmitter would be able to see howthe signals are faring and might want to make adjustments in the powersallocated to its antennas. Perhaps a smart computer at the transmitterwill decide to not transmit on one antenna, if the received signals areso much smaller (in amplitude) than the other two antennas. Maybe thepower should be split between antenna 2 and 3 and turn off antenna 1until the channel improves.

Modeling a MIMO Channel

Starting with a general channel which has both multipath and Doppler(the conditions facing a mobile in case of a cell phone system), thechannel matrix H for this channel takes this form.

${H\left( {\tau,t} \right)} = \begin{bmatrix}{\mspace{34mu} {h_{11}\left( {\tau,t} \right)}} & {{h_{11}\left( {\tau,t} \right)}\mspace{14mu} \cdots} & {h_{1N_{T}}\left( {\tau,t} \right)} \\{\mspace{34mu} {h_{21}\left( {\tau,t} \right)}} & {{h_{22}\left( {\tau,t} \right)}\mspace{14mu} \cdots} & {h_{2N_{T}}\left( {\tau,t} \right)} \\{\mspace{45mu} \vdots} & {\vdots \mspace{79mu} \ddots} & \vdots \\{h_{N_{R,}1}\mspace{14mu} \left( {\tau,t} \right)} & {{h_{N_{R,}2}\left( {\tau,t} \right)}\mspace{14mu} \ldots \mspace{14mu} h_{N_{R,}N_{T}}} & \left( {\tau,t} \right)\end{bmatrix}$

Each path coefficient is a function of not only time t because thetransmitter is moving but also a time delay relative to other paths. Thevariable τ indicates relative delays between each component caused byfrequency shifts. The time variable t represents the time-varying natureof the channel such as one that has Doppler or other time variations.

If the transmitted signal is s_(i)(t), and the received signal isr_(i)(t), the input-output relationship of a general MIMO channel isdefined as:

$\begin{matrix}{{r_{i}(t)} = {\sum\limits_{j = 1}^{N_{T}}\; {\int_{- \infty}^{\infty}{{h_{ij}\left( {\tau,t} \right)}\mspace{14mu} {S_{j}\left( {t - \tau} \right)}{dt}}}}} \\{{= {{\sum\limits_{j = 1}^{N_{T}}\; {{h_{ij}\left( {\tau,t} \right)}*{S_{j}(\tau)}\mspace{14mu} i}} = 1}},{2\mspace{14mu} \ldots \mspace{14mu} N_{R}}}\end{matrix}$

The channel equation for the received signal r_(i)(t) is expressed as aconvolution of the channel matrix H and the transmitted signals becauseof the delay variable τ. This relationship can be defined in matrix formas:

r(t)=H(τ,t)*s(t)

If the channel is assumed to be flat (non-frequency selective), but istime-varying, i.e. has Doppler, the relationship is written without theconvolution as:

r(t)=H(t)s(t)

In this case, the H matrix changes randomly with time. If the timevariations are very slow (non-moving receiver and transmitter) such thatduring a block of transmission longer than the several symbols, thechannel can be assumed to be non-varying, or static. A fixed realizationof the H matrix for a fixed Point-to-Point scenario can be written as:

r(t)=H(t)s(t)

The individual entries can be either scalar or complex.

For analysis purposes, important assumptions can be made about the Hmatrix. We can assume that it is fixed for a period of one or moresymbols and then changes randomly. This is a fast change and causes theSNR of the received signal to change very rapidly. Or we can assume thatit is fixed for a block of time, such as over a full code sequence,which makes decoding easier because the decoder does not have to dealwith a variable SNR over a block. Or we can assume that the channel issemi-static such as in a TDMA system, and its behavior is static over aburst or more. Each version of the H matrix seen is a realization. Howfast these realizations change depends on the channel type.

$H = {\begin{bmatrix}{h_{11}\mspace{25mu}} & {h_{12}\mspace{14mu} \cdots} & h_{1N_{T}} \\h_{21} & {h_{22}\mspace{45mu} \cdots} & h_{2N_{T}} \\{\mspace{20mu} \vdots} & {\vdots \mspace{20mu} \ddots} & \vdots \\h_{N_{R},1} & {h_{N_{R,}2}\mspace{14mu} \ldots \mspace{14mu} h_{N_{R,}1}} & \;\end{bmatrix}\quad}$

For a fixed random realization of the H matrix, the input-outputrelationship can be written without the convolution as:

r(t)=H s(t)

In this channel model, the H matrix is assumed to be fixed. An exampleof this type of situation where the H matrix may remain fixed for a longperiod would be a Point-to-Point system where we have fixed transmitterand receiver. In most cases, the channel can be considered to be static.This allows us to treat the channel as deterministic over that periodand amenable to analysis. In a point-to-point system, the channel issemi-static and it behavior is static over a burst or more. Each versionof the H matrix is a realization.

The power received at all receive antennas is equal to the sum of thetotal transmit power, assuming channel offers no gain or loss. Eachentry h_(ij) comprises an amplitude and phase term. Squaring the entryh_(ij) give the power for that path. There are N_(T) paths to eachreceiver, so the sum of j terms, provides the total transmit power. Eachreceiver receives the total transmit power. For this relation, thetransmit power of each transmitter is assumed to be 1.

${\sum\limits_{j = 1}^{N_{T}}\; \left( h_{ij} \right)^{2}} = {{N_{T}\mspace{25mu} (1)} = N_{T}}$

The H matrix is a very important construct in understanding MIMOcapacity and performance. How a MIMO system performs depends on thecondition of the channel matrix H and its properties. The H matrix canbe thought of as a set of simultaneous equations. Each equationrepresents a received signal which is a composite of unique set ofchannel coefficients applied to the transmitted signal.

r ₁ =h ₁₁ s+h ₁₂ s ^(. . .) +h _(1N) ₂ s

If the number of transmitters is equal to the number of receivers, thereexists a unique solution to these equations. If the number of equationsis larger than the number of unknowns (i.e. N_(R)>N_(T)), the solutioncan be found using a zero-forcing algorithm. When N_(T)=N_(R), (thenumber of transmitters and receivers are the same), the solution can befound by (ignoring noise) inverting the H matrix as in:

ŝ(t)=H ⁻¹ r(t)

The system performs best when the H matrix is full rank, with eachrow/column meeting conditions of independence. What this means is thatbest performance is achieved only when each path is fully independent ofall others. This can happen only in an environment that offers richscattering, fading and multipath, which seems like a counter-intuitivestatement. Looking at the equation above, the only way to extract thetransmitted information is when the H matrix is invertible. And the onlyway it is invertible is if all its rows and columns are uncorrelated.And the only way this can occur is if the scattering, fading and allother effects cause the channels to be completely uncorrelated.

Diversity Domains and MIMO Systems

In order to provide a fixed quality of service, a large amount oftransmit power is required in a Rayleigh or Rican fading environment toassure that no matter what the fade level, adequate power is stillavailable to decode the signal. Diversity techniques that mitigatemultipath fading, both slow and fast are called Micro-diversity, whereasthose resulting from path loss, from shadowing due to buildings etc. arean order of magnitude slower than multipath, are called Macro-diversitytechniques. MIMO design issues are limited only to micro-diversity.Macro-diversity is usually handled by providing overlapping base stationcoverage and handover algorithms and is a separate independentoperational issue.

In time domain, repeating a symbol N times is the simplest example ofincreasing diversity. Interleaving is another example of time diversitywhere symbols are artificially separated in time so as to createtime-separated and hence independent fading channels for adjacentsymbols. Error correction coding also accomplishes time-domain diversityby spreading the symbols in time. Such time domain diversity methods aretermed here as Temporal diversity.

Frequency diversity can be provided by spreading the data overfrequency, such as is done by spread spectrum systems. In OFDM frequencydiversity is provided by sending each symbol over a different frequency.In all such frequency diversity systems, the frequency separation mustbe greater than the coherence bandwidth of the channel in order toassure independence.

The type of diversity exploited in MIMO is called Spatial diversity. Thereceive side diversity, is the use of more than one receive antenna. SNRgain is realized from the multiple copies received (because the SNR isadditive). Various types of linear combining techniques can take thereceived signals and use special combining techniques such are MaximalRatio Combining, Threshold Combing etc. The SNR increase is possible viacombining results in a power gain. The SNR gain is called the arraygain.

Transmit side diversity similarly means having multiple transmitantennas on the transmit side which create multiple paths and potentialfor angular diversity. Angular diversity can be understood asbeam-forming. If the transmitter has information about the channel, asto where the fading is and which paths (hence direction) is best, thenit can concentrate its power in a particular direction. This is anadditional form of gain possible with MIMO.

Another form of diversity is Polarization diversity such as used insatellite communications, where independent signals are transmitted oneach polarization (horizontal vs. vertical). The channels, although atthe same frequency, contain independent data on the two polarized henceorthogonal paths. This is also a form of MIMO where the two independentchannels create data rate enhancement instead of diversity. So satellitecommunications is a form of a (2, 2) MIMO link.

Related to MIMO but not MIMO

There are some items that need be explored as they relate to MIMO butare usually not part of it. First are the smart antennas used in set-topboxes. Smart antennas are a way to enhance the receive gain of a SISOchannel but are different in concept than MIMO. Smart antennas usephased-arrays to track the signal. They are capable of determining thedirection of arrival of the signal and use special algorithms such asMUSIC and MATRIX to calculate weights for its phased arrays. They areperforming receive side processing only, using linear or non-linearcombining.

Rake receivers are a similar idea, used for multipath channels. They area SISO channel application designed to enhance the received SNR byprocessing the received signal along several “fingers” or correlatorspointed at particular multipath. This can often enhance the receivedsignal SNR and improve decoding. In MIMO systems Rake receivers are notnecessary because MIMO can actually simplify receiver signal processing.

Beamforming is used in MIMO but is not the whole picture of MIMO. It isa method of creating a custom radiation pattern based on channelknowledge that provides antenna gains in a specific direction. Beamforming can be used in MIMO to provide further gains when thetransmitter has information about the channel and receiver locations.

Importance of Channel State Information

Dealing with Channel matrix H is at the heart of how MIMO works. Ingeneral, the receiver is assumed to be able to get the channelinformation easily and continuously. It is not equally feasible for thetransmitter to obtain a fresh version of the channel state information,because the information has gotten impaired on the way back. However, aslong as the transit delay is less than channel coherence time, theinformation sent back by the receiver to the transmitter retains itsfreshness and usefulness to the transmitter in managing its power. Atthe receiver, we refer to channel information as Channel State (or side)Information at the Receiver, CSIR. Similarly when channel information isavailable at the transmitter, it is called CSIT. CSI, the channel matrixcan be assumed to be known instantaneously at the receiver or thetransmitter or both. Although in short term the channel can have anon-zero mean, it is assumed to be zero-mean and uncorrelated on allpaths. When the paths are correlated, then clearly, less information isavailable to exploit. But the channel can still be made to work.

Channel information can be extracted by monitoring the received gains ofa known sequence. In Time Division Duplex (TDD) communications whereboth transmitter and the receiver are on the same frequency, the channelcondition is readily available to the transmitter. In Frequency DivisionDuplex (FDD) communications, since the forward and reverse links are atdifferent frequencies, this requires a special feedback link from thereceiver to the transmitter. In fact receive diversity alone is veryeffective but it places greater burden on the smaller receivers,requiring larger weight, size and complex signal processing henceincreasing cost.

Transmit diversity is easier to implement in a cellular system from asystem point of view because the base station towers in a cell systemare not limited by power or weight. In addition to adding more transmitantennas on the base station towers, space-time coding is also used bythe transmitters. This makes the signal processing required at thereceiver simpler.

MIMO Gains

Our goal is to transmit and receive data over several independentlyfading channels such that the composite performance mitigates deep fadeson any of the channels. To see how MIMO enhances performance in a fadingor multipath channel, the BER for a BPSK signal is examined as afunction of the receive SNR.

P _(e) ≈Q(√{square root over (2∥h∥ ²SNR)})

The quantity (h²×SNR) is the instantaneous SNR.

Now assume that there are L possible paths, where L=N_(R)×N_(T), withN_(T)=number of transmitter and N_(R)=number of receive antennas. Sincethere are several paths, the average BER can be expressed as a functionof the average channel gain over all these paths. This quantity is theaverage gain over all channels, L.

$\left. ||h \right.||^{2} = {{{Avg}\left\lbrack \left| h_{1} \right|^{2} \right\rbrack} = {\sum\limits_{l = 1}^{L}\; \left| h_{1} \right|^{2}}}$

The average SNR can be rewritten as a product of two terms.

$\left. ||h||{}_{2}{SNR} \right. = \left. {\underset{\_}{L \times {SNR}} \cdot \frac{1}{L}}||h \right.||^{2}$

The first part (L×SNR) is a linear increase in SNR due to the L paths.This term is called by various names, including power gain, rate gain orarray gain. This term can also include beamforming gain. Henceincreasing the number of antennas increases the array gain directly bythe factor L. The second term is called diversity gain. This is theaverage gain over L different paths.It seems intuitive that if one of the paths exhibits deep fading then,when averaged over a number of independent paths, the deep fades can beaveraged out. (We use the term channel to mean the composite of allpaths.) Hence on the average we would experience a diversity gain aslong as the path gains across the channels are not correlated. If thegains are correlated, such as if all paths are mostly line-of-sight, wewould obtain only an array gain and very little diversity gain. This isintuitive because a diversity gain can come only if the paths arediverse, or in other words uncorrelated.

MIMO Advantages

Operating in Fading Channels

The most challenging issue in communications signal design is how tomitigate the effects of fading channels on the signal BER. A fadingchannel is one where channel gain is changing dramatically, even at highSNR, and as such it results in poor BER performance as compared to anAWGN channel. For communications in a fading channel, a way to convertthe highly variable fading channel to a stable AWGN-like channel isneeded.

Multipath fading is a phenomenon that occurs due to reflectors andscatters in the transmission path. The measure of multipath is DelaySpread, which is the RMS time delay as a function of the power of themultipath. This delay is converted to a Coherence Bandwidth (CB), ametric of multipath. A time delay is equivalent to a frequency shift inthe frequency domain. So any distortion that delays a signal, changesits frequency. Thus, delay spread >bandwidth distortion.

Whether a signal is going through a flat or a frequency-selective fadingat any particular time is a function of coherence bandwidth of thechannel as compared with its bandwidth as shown in Table I. If theCoherence bandwidth of the channel is larger than the signal bandwidth,then we have a flat or a non-frequency selective channel. What coherencemeans is that all the frequencies in the signal respond similarly or aresubject to the same amplitude distortion. This means that fading doesnot affect frequencies differentially, which is a good thing.Differential distortion is hard to deal with. So of all types of fading,flat fading is the least problematic.

The next source of distortion is Doppler. Doppler results in differentdistortions to the frequency band of the signal. The measure of Dopplerspread is called Coherence Time (CT) (no relationship to CoherenceBandwidth from the flat-fading case). The comparison of the CT with thesymbol time determines the speed of fading. So if the coherence time isvery small, compared to the symbol time, that's not good.

The idea of Coherence Time and Coherence Bandwidth is often confused.Flatness refers to frequency response and not to time. So CoherenceBandwidth determines whether a channel is considered flat or not.

Coherence Time, on the other hand has to do with changes over time,which is related to motion. Coherence Time is the duration during whicha channel appears to be unchanging. One can think about Coherence Timewhen Doppler or motion is present. When Coherence Time is longer thansymbol time, then a slow fading channel is provided and when symbol timeis longer than Coherence time, a fast fading channel is provided. Soslowness and fastness mean time based fading.

TABLE I MIMO channel Types and their Measures Channel Channel SpreadSelectivity Type Measure Delay Spread Frequency Non-selective CoherenceBandwidth > Signal Bandwidth Frequency Selective Signal Bandwidth >Coherence Bandwidth Doppler Spread Time Slow-fading Coherence Time >Symbol Time Fast-fast Symbol Time > fading Coherence Time Angle SpreadBeam pattern — Coherence Distance

In addition to these fast channel effects, there are mean path losses aswell as rain losses, which are considered order-of-magnitude slowereffects and are managed operationally and so will not be part of theadvantages of MIMO.

How MIMO Creates Performance Gains in a Fading Channel

Shannon defines capacity of a channel as a function of its SNR.Underlying this is the assumption that the SNR is invariant. For such asystem, Shannon capacity is called its ergodic capacity. Since SNR isrelated to BER, the capacity of a channel is directly related to howfast the BER declines with SNR. The BER needs to decrease quickly withincreasing power.

The Rayleigh channel BER when compared to an AWGN channel for the sameSNR is considerably bigger and hence the capacity of a Rayleigh channelwhich is the converse of its BER, is much lower. Using the BER of a BPSKsignal as a benchmark, one can examine the shortfall of a Rayleighchannel and see how MIMO can help to mitigate this loss.

For a BPSK signal, the BER in an AWGN channel is given by setting∥h²∥=1.

P _(e) ≈Q(√{square root over (2SNR)})

The BER of the same channel in a Rayleigh channel is given by

$P_{e} = {{\frac{1}{2}\mspace{14mu} \left( {1 - \sqrt{\frac{SNR}{1 + {SNR}}}} \right)} \approx \frac{1}{2{SNR}}}$

FIG. 151 shows the BER of an AWGN and a Rayleigh channel as a functionof the SNR.

The AWGN BER 15102 varies by the inverse of the square of the SNR, SNR⁻²and declines much faster than the Rayleigh channel 15104 which declinesinstead by SNR⁻¹. Hence an increase in SNR helps the Rayleigh channel15104 much less than it does an AWGN channel 15102. The Rayleigh channel15104 improves much more slowly as more power is added.

FIG. 151 illustrates that for a BER of 10⁻³, an additional 17 dB ofpower is required in a Rayleigh channel 15104. This is a very largedifferential, nearly 50 times larger than AWGN 15102. One way to bringthe Rayleigh curve 15104 closer to the AWGN curve 15102 (which forms alimit of performance) is to add more antennas on the receive or thetransmit-side hence making SISO into a MIMO system.

Starting with just one antenna, the number of receive antennas isincreased to N_(R), while keeping one transmitter, making it a SIMOsystem. By looking at the asymptotic BER at large SNR (large SNR has noformal definition, anything over 15 dB can be considered large), adetermination can be made of the gain caused by adding just one moreantenna to the N antennas. The ratio of the BER to the BER due to onemore antenna may then be determined.

$\frac{{BER}\left( {N,{SNR}} \right)}{{BER}\left( {{N + 1},{SNR}} \right)} = {{SNR}\mspace{14mu} \left( {1 + \frac{1}{{2N} + 1}} \right)}$

The gain from adding one more antenna is equal to SNR multiplied by adelta increase in SNR. The delta increase diminishes as more and moreantennas are added. The largest gain is seen when going from a singleantenna to two antennas, (1.5 for going from 1 to 2 vs. 1.1 for goingfrom 4 to 5 antennas). This delta increase is similar in magnitude tothe slope of the BER curve at large SNR.

Formally, a parameter called Diversity order d, is defined as the slopeof the BER curve as a function of SNR in the region of high SNR.

$d = {- {\lim\limits_{{SNR}\rightarrow\infty}\mspace{14mu} {\log \mspace{14mu} \frac{{BER}({SNR})}{SNR}}}}$

FIG. 152 shows the gains possible with MIMO as more receive antennas areadded. As more and more antennas are added, a Rayleigh channelapproaches the AWGN channel. Can one keep on increasing the number ofantennas indefinitely? No, beyond a certain number, increase in numberof paths L does not lead to significant gains. When complexity is takeninto account, a small number of antennas is enough for satisfactoryperformance.

Capacity of MIMO Channels Capacity of a SISO Channel

All system designs strive for a target capacity of throughput. For SISOchannels, the capacity is calculated using the well-known Shannonequation. Shannon defines capacity for an ergodic channel that data ratewhich can be transmitted with asymptotically small probability of error.The capacity of such a channel is given in terms of bits/sec or bynormalizing with bandwidth by bits/sec/Hz. The second formulation allowseasier comparison and is the one used more often. It is also bandwidthindependent.

$C = {W\; \log_{2}\mspace{14mu} \left( {1 + \frac{P}{N_{o}W}} \right)\mspace{14mu} b\text{/}s}$$\frac{C}{W} = {\log_{2}\mspace{14mu} \left( {1 + {SNR}} \right)\mspace{14mu} b\text{/}s\mspace{14mu} \text{/}H_{z}}$

At high SNRs, ignoring the addition of 1 to SNR, the capacity is adirect function of SNR.

$\frac{C}{W} \approx {\log_{2}\mspace{14mu} ({SNR})\mspace{14mu} b\text{/}s\mspace{14mu} \text{/}H_{z}}$

This capacity is based on a constant data rate and is not a function ofwhether channel state information is available to the receiver or thetransmitter. This result is applicable only to ergodic channels, oneswhere the data rate is fixed and SNR is stable.

Capacity of MIMO Channels

Shannon's equation illustrates that a particular SNR can give only afixed maximum capacity. If SNR goes down, so will the ability of thechannel to pass data. In a fading channel, the SNR is constantlychanging. As the rate of fade changes, the capacity changes with it.

A fixed H matrix can be used as a benchmark of performance where thebasic assumption is that, for that one realization, the channel is fixedand hence has an ergodic channel capacity. In other words, for just thatlittle time period, the channel is behaving like an AWGN channel. Bybreaking a channel into portions of either time or frequency so that insmall segments, even in a frequency-selective channel with Doppler,channel can be treated as having a fixed realization of the H matrix,i.e. allowing us to think of it instantaneously as a AWGN channel. Thecapacity calculations are preformed over several realizations of Hmatrix and then compute average capacity over these. In flat fadingchannels the channel matrix may remain constant and or may change veryslowly. However, with user motion, this assumption does not hold.

Decomposing a MIMO Channel into Parallel Independent Channels

Conceptually MIMO may be thought of as the transmission of same dataover multiple antennas, hence it is a matrix channel. But there is amathematical trick that lets us decompose the MIMO channel into severalindependent parallel channels each of which can be thought of as a SISOchannel. To look at a MIMO channel as a set of independent channels, analgorithm called Singular Value Decomposition (SVD) is used. The processrequires pre-coding at the transmitter and receiver shaping at thereceiver.

Input and Output Auto-Correlation

Assuming that a MIMO channel has N transmitters and M receivers, thetransmitted vector across NT antennas is given by x₁, x₂, x₃, . . .x_(NT). Individual transmit signals consist of symbols that are zeromean circular-symmetric complex Gaussian variables. (A vector x is saidto be circular-symmetric if e^(j⊖)x has same distribution for all ⊖.)The covariance matrix for the transmitted symbols is written as:

R _(xx) =E{xx ^(H)}

Where symbol H stands for the transpose and component-wise complexconjugate of the matrix (also called Hermitian) and not the channelmatrix. This relationship gives us a measure of correlated-ness of thetransmitted signal amplitudes.

When the powers of the transmitted symbols are the same, a scaledidentity matrix is provided. For a (3×3) MIMO system of total power ofP_(T), equally distributed this matrix is written as:

$R_{xx} = {P_{T}\mspace{14mu}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}$

If the same system distributes the power differently say in ratio of1:2:3, then the covariance matrix would be:

$R_{xx} = {P_{T}\mspace{14mu}\begin{bmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{bmatrix}}$

If the total transmitted power is P_(T) and is equal to trace of theInput covariance matrix, the total power of the transmitted signal canbe written as the trace of the covariance matrix.

P=tr{R _(xx)}

The received signal is given by:

r=Hx+n

The noise matrix (N×1) components are assumed to be ZMGV (zero-meanGaussian variable) of equal variance. We can write the covariance matrixof the noise process similar to the transmit symbols as:

R _(nn) =E{nn ^(H)}

And since there is no correlation between its rows, this can be writtenas:

R _(nn)=σ²1_(M)

Which says that each of the M received noise signals is an independentsignal of noise variance, σ².

Each receiver receives a complex signal consisting of the sum of thereplicas from N transmit antennas and an independent noise signal.Assuming that the power received at each receiver is not the same, theSNR of the m_(th) receiver may be written as:

$\gamma_{m} = \frac{P_{m}}{\sigma^{2}}$

where P_(m) is some part of the total power. However the average SNR forall receive antennas would still be equal to P_(T)/σ², where P_(T) istotal power because

$P_{T} = {\sum\limits_{m = 1}^{N_{T}}\; P_{m}}$

Now we write the covariance matrix of the receive signal using as:

R _(rr) =HR _(xx) H ^(H) +R _(nn)

where R_(xx) is the covariance matrix of the transmitted signal. Thetotal receive power is equal to the trace of the matrix R_(rr).

Singular Value Decomposition (SVD)

Referring now to FIG. 153 there is illustrated a modal decomposition ofa MIMO channel with full CSI. SVD is a mathematical application thatlets us create an alternate structure of the MIMO signal. In particular,the MIMO signal is examined by looking at the eigenvalues of the Hmatrix. The H matrix can be written in Singular Value Decomposition(SVD) form as:

H=UΣV ^(H)

where U and V are unitary matrices (U^(H)U=I_(NR), and V^(H)V=I_(NT))and Σ is a N_(R)×N_(T) diagonal matrix of singular values (

) of H matrix. If H is a full Rank matrix then we have a min (N_(R),N_(T)) of non-zero singular values, hence the same number of independentchannels. The parallel decomposition is essentially a linear mappingfunction performed by pre-coding the input signal by multiplying it withmatrix V.

x=V{tilde over (x)}

The received signal y is given by multiplying it with U^(H),

{tilde over (y)}=U ^(H)(Hx+n)

Now multiplying it out, and setting value of H, to get:

{tilde over (y)}=U ^(H)(UΣV ^(H) x+n)

Now substitute:

x=V{tilde over (x)}

To obtain:

$\begin{matrix}{\overset{\sim}{\gamma} = {U^{H}\mspace{14mu} \left( {{\underset{\_}{U\; \Sigma \; V^{H}}x} + n} \right)}} \\{= {U^{H}\mspace{14mu} \left( {{U\; \Sigma \; V^{\dagger}V\overset{\sim}{x}} + n} \right)}} \\{= {{U^{H}\mspace{14mu} U\; \Sigma \; V^{\dagger}V\overset{\sim}{x}} + {U^{H}\mspace{14mu} n}}} \\{= {{\Sigma \; \overset{\sim}{x}} + \overset{\sim}{n}}}\end{matrix}$

In the last result, the output signal is in form of a pre-coded inputsignal times the singular value matrix, Σ. Note that the multiplicationof noise n, by the unitary matrix U^(H) does not change the noisedistribution. Note that the only way SVD can be used is if thetransmitter knows what pre-coding to apply, which of course requiresknowledge of the channel by the transmitter. As shown in FIG. 154, SVDis used for decomposing a matrix channel is decomposed into parallelequivalent channels.

Since SVD entails greater complexity, not the least of which is feedingback CSI to the transmitter, with the same results, why should weconsider the SVD approach? The answer is that the SVD approach allowsthe transmitter to optimize its distribution of transmitted power,thereby providing a further benefit of transmit array gain.

The channel eigenmodes (or principle components) can be viewed asindividual channels characterized by coefficients (eigenvalues). Thenumber of significant eigenvalues specifies the maximum degree ofdiversity. The larger a particular eigenvalue, the more reliable is thatchannel. The principle eigenvalue specifies the maximum possiblebeamforming gain. The most important benefit of the SVD approach is thatit allows for enhanced array gain—the transmitter can send more powerover the better channels, and less (or no) power over the worst ones.The number of principle components is a measure of the maximum degree ofdiversity that can be realized in this way.

Channel Capacity of a SIMO, MISO Channel

Before discussing the capacity of a MIMO channel, let's examine thecapacity of a channel that has multiple receivers or transmitters butnot both. We modify the SNR of a SISO channel by the gain factorobtained from having multiple receivers.

C _(SIMO)=log₂(1+∥h∥ ²SNR)bits/s/Hz

The channel consists of only N_(R) paths and hence the channel gain isconstrained by:

∥h∥ ² =N _(R)

This gives the ergodic capacity of the SIMO channel as:

C _(SIMO)=log₂(1+N _(R)SNR)bits/s/Hz

So the SNR is increasing by a factor of N_(R). This is a logarithmicgain. Note that we are assuming that the transmitter has no knowledge ofthe channel.

Next consider a MISO channel, with multiple transmitters but onereceiver. The channel capacity of a MISO channel is given by:

$C_{MISO} = {\log_{2}\mspace{14mu} \left( {1 + \frac{\left. ||h||{}_{2}\mspace{14mu} {SNR} \right.}{N_{T}}} \right)\mspace{14mu} {bits}\text{/}s\text{/}{Hz}}$

Why divide by N_(T)? Compared to the SIMO case, where each path has SNRbased on total power, in this case, total power is divided by the numberof transmitters. So the SNR at the one receiver keeps getting smaller asmore and more transmitters are added. For a two receiver case, each pathhas a half of the total power. But since there is only one receiver,this is being divided by the total noise power at the receiver, so theSNR is effectively cut in half

Again if the transmitter has no knowledge of the channel, the equationdevolves in to a SISO channel:

C _(MISO)=log₂(1+SNR)bits/s/Hz

The capacity of a MISO channel is less than a SIMO channel when thechannel in unknown at the transmitter. However, if the channel is knownto the transmitter, then it can concentrate its power into one channeland the capacity of SIMO and MISO channel becomes equal under thiscondition.

Both SIMO and MISO can achieve diversity but they cannot achieve anymultiplexing gains. This is obvious for the case of one transmitter,(SIMO). In a MISO system all transmitters would need to send the samesymbol because a single receiver would have no way of separating thedifferent symbols from the multiple transmitters. The capacity stillincreases only logarithmically with each increase in the number of thetransmitters or the receivers. The capacity for the SIMO and MISO arethe same. Both channels experience array gain of the same amount butfall short of the MIMO gains.

Assuming a discrete MIMO channel model as shown in FIG. 155. The channelgain may be time-varying but assumed to be fixed for a block of time andrandom. Assume that total transmit power is P, bandwidth is B and thePSD of noise process is N₀/2.

Assume that total power is limited by the relationship:

${E\mspace{14mu} \left( {x^{H}\mspace{14mu} x} \right)} = {{\sum\limits_{i = 1}^{N_{T}}\; {E\mspace{14mu} \left\{ \left| x_{i} \right|^{2} \right\}}} = N_{T}}$

We write The input covariance matrix may be written as R_(xx)

R _(xx) =E{xx ^(H)}

The trace of this matrix is equal to:

$\rho = {\frac{P}{\# \; {paths}} = {{tr}\mspace{14mu} \left\{ R_{xx} \right\}}}$

or the power per path. When the powers are uniformly distributed (equal)then this is equal to a unity matrix. The covariance matrix of theoutput signal would not be unity as it is a function of the H matrix.

Now the capacity expression for a MIMO matrix channel using a fixed butrandom realization the H matrix can be developed. Assuming availabilityof CSIR, the capacity of a deterministic channel is defined by Shannonas:

$C = {\max\limits_{f\mspace{14mu} {(x)}}\mspace{14mu} {I\left( {x;y} \right)}}$

I(x;y) is called the mutual information of x and y. It is the capacityof the channel is the maximum information that can be transmitted from xto y by varying the channel PDF. The value f(x) is the probabilitydensity function of the transmit signal x. From information theory, therelationship of mutual information between two random variables as afunction of their differential entropy may be obtained.

I(x;y)=ε_((y))−ε_((y|x))

The second term is constant for a deterministic channel because it isfunction only of the noise. So mutual information is maximum only whenthe term H(y), called differential entropy is maximum.

The differential entropy H(y) is maximized when both x and y arezero-mean, Circular-Symmetric Complex Gaussian (ZMCSCG) random variable.Also from information theory, the following relationships are provided:

ε_((y))=log₂ {det(πeR _(yy)}

ε_((y|x))=log₂ {det(πeN _(O) I _(N) _(R) }

Now we write the signal y as:

y=√{square root over (γ)}Hx+z

Here γ is instantaneous SNR. The auto-correlation of the output signal ywhich we need for (27.48) is given by

$\begin{matrix}{R_{yy} = {E\left\{ {yy}^{H} \right\}}} \\{= {E\left\{ {\left( {{\sqrt{\gamma}{Hx}} + z} \right)\left( {{\sqrt{\gamma}x^{H}H^{H}} + z^{H}} \right)} \right\}}} \\{= {E\left\{ \left( {{\gamma \; {Hxx}^{H}H^{H}} + {zz}^{H}} \right) \right\}}} \\{= {{\gamma \; {HE}\left\{ {xx}^{H} \right\} H^{H}} + {E\left\{ {zz}^{H} \right\}}}}\end{matrix}$ R_(yy) = γ HR_(xx)H^(H) + N_(o)I_(N_(R))

From here we can write the expression for capacity as

$C = {\log_{2}\mspace{14mu} \det \mspace{14mu} \left( {I_{N_{R}} + {\frac{SNR}{N_{T}}\mspace{14mu} {HH}^{H}}} \right)}$

When CSIT is not available, we can assume equal power distribution amongthe transmitters, in which case R_(xx) is an identity matrix and theequation becomes

$C = {\log_{2}\mspace{14mu} \det \mspace{14mu} \left( {I_{N_{R}} + {\frac{SNR}{N_{T}}\mspace{14mu} {HH}^{H}}} \right)}$

This is the capacity equation for MIMO channels with equal power. Theoptimization of this expression depends on whether or not the CSI (Hmatrix) is known to the transmitter.

Now note that as the number of antennas increases, we get

${\lim\limits_{N\rightarrow\infty}{\frac{I}{M}\mspace{14mu} {HH}^{H}}} = I_{N}$

This means that as the number of paths goes to infinity, the power thatreaches each of the infinite number of receivers becomes equal and thechannel now approaches an AWGN channel.

This gives us an expression about the capacity limit of a N_(T)×N_(R)MIMO system.

C=M log₂ det(I _(N) _(R) +SNR)

where M is the minimum of N_(T) and N_(R), the number of the antennas.Thus, the capacity increases linearly with M, the minimum of (N_(T),N_(R)). If a system has (4, 6) antennas, the maximum diversity that canbe obtained is of order 4, the small number of the two systemparameters.

Channel Known at Transmitter

The SVD results can be used to determine how to allocate powers acrossthe transmitters to get maximum capacity. By allocating the powernon-equally, the capacity can be increased. In general, channels withhigh SNR (high

), should get more power than those with lower SNR.

There is a solution to the power allocation problem at the transmittercalled the water-filling algorithm. This solution is given by:

$\frac{P_{i}}{P} = \left\{ \begin{matrix}\left( {\frac{I}{\gamma_{o}} - \frac{I}{\gamma_{i}}} \right) & {\gamma_{i} > \gamma_{o}} \\{O} & {\gamma_{i} \leq \gamma_{o}}\end{matrix} \right.$

Where γ₀ is a threshold constant. Here γ_(I) is the SNR of the i_(th)channel.

When comparing the inverse of the threshold with the inverse of thechannel SNR and the inverse difference is less than the threshold, nopower is allocated to the i_(th) channel. If the difference is positivethen more power may be added to see if it helps the overall performance.

The capacity using the water-filling algorithm is given by:

$C = {\sum\limits_{{i\text{:}\gamma_{i}} > \gamma_{o}}{B\mspace{14mu} \log_{2}\mspace{14mu} \left( \frac{\gamma_{i}}{\gamma_{o}} \right)}}$

The thing about the water-filling algorithm is that it is much easier tocomprehend then is it to describe using equations. Think of it as a boatsinking in the water. Where would a person sit on the boat while waitingfor rescue, clearly the part that is sticking above the water, right?The analogous part to the boat above the surface are the channels thatcan overcome fading. Some of the channels reach the receiver with enoughSNR for decoding. So the data/power should go to these channels and notto the ones that are under water. So basically, power is allocated tothose channels that are strongest or above a pre-set threshold.

Channel Capacity in Outage

The Rayleigh channels go through such extremes of SNR fades that theaverage SNR cannot be maintained from one time block to the next. Due tothis, they are unable to support a constant data rate. A Rayleighchannel can be characterized as a binary state channel; an ergodicchannel but with an outage probability. When it has a SNR that is abovea minimum threshold, it can be treated as ON and capacity can becalculated using the information-theoretic rate. But when the SNR isbelow the threshold, the capacity of the channel is zero. The channel issaid to be in outage.

Although ergodic capacity can be useful in characterizing a fast-fadingchannel, it is not very useful for slow-fading, where there can beoutages for significant time intervals. When there is an outage, thechannel is so poor that there is no scheme able to communicate reliablyat a certain fixed data rate.

The outage capacity is the capacity that is guaranteed with a certainlevel of reliability. The outage capacity is defined as the informationrate that is guaranteed for (100−p) % of the channel realizations. A 1%outage probability means that 99% of the time the channel is above athreshold of SNR and can transmit data. For real systems, outagecapacity is the most useful measure of throughput capability.

Question: which would have higher capacity, a system with 1% outage or10% outage? The high probability of outage means that the threshold canbe set lower, which also means that system will have higher capacity, ofcourse only while it is working which is 90% of the time.

The capacity equation of a Rayleigh channel with outage probability ε,can be written as:

C _(out=(1−P) _(out) _()Blog 2)(1+γ_(min))

The probability of obtaining a minimum threshold value of the SNR,assuming it has a Rayleigh distribution, can be calculated. The capacityof channel under outage probability ε is given by:

$C_{ɛ} = {\log_{2}\mspace{14mu} \left( {1 + {{{SNR} \cdot \ln}\mspace{14mu} \left( \frac{1}{1 - ɛ} \right)}} \right)}$

The Shannon's equation has been modified by the outage probability.

Capacity Under a Correlated Channel

MIMO gains come from the independence of the channels. The developmentof ergodic capacity assumes that channels created by MIMO areindependent. But what happens if there is some correlation among thechannels which is what happens in reality due to reflectors located nearthe base station or the towers. Usually in cell phone systems, thetransmitters (on account on being located high on towers) are lesssubject to correlation than are the receivers (the cell phones).

The signal correlation, r, between two antennas located a distance dapart, transmitting at the same frequency, is given by zero order Besselfunction defined as:

$r = {J_{o}^{2}\mspace{14mu} \left( \frac{2\pi \; d}{\lambda} \right)}$

where J₀(x) is the zero-th order Bessel function. FIG. 156 shows thecorrelation coefficient r, plotted between receive antennas vs. d/λ,using the Jakes model.

An antenna that is approximately half a wavelength away experiences only10% correlation with the first. To examine the effect that correlationhas on system capacity, the channel mat channel matrix H is replaced inthe ergodic equation as follows:

$C = {\log_{2}\mspace{14mu} \det \mspace{14mu} \left( {I_{N_{R}} + {\frac{SNR}{N_{T}}\mspace{14mu} {HH}^{H}}} \right)}$

Assuming equal transmit powers, with a correlation matrix, assuming thatthe following normalization holds. This normalization allows thecorrelation matrix, rather than covariance.

${\sum\limits_{i,{j = 1}}^{N_{T},N_{R}}\; \left| h_{i,j} \right|^{2}} = 1$

Now we write the capacity equation instead as

$C = {\log_{2}\mspace{14mu} M\; \det \mspace{14mu} \left( {I + {\frac{SNR}{M} \cdot R}} \right)}$

where R is the normalized correlation matrix, such that its components

${r_{{ij} = \frac{1}{\sqrt{\sigma_{i}\sigma_{j}}}}{\sum\limits_{K}\mspace{14mu} {h_{ik}h_{jk}^{*}}}} = {\sum\limits_{K}\mspace{14mu} {h_{ik}h_{jk}^{*}}}$

We can write the capacity equation as:

C=M·log₂ det(1+SNR)+log₂ det(R)

The first underlined part of the expression is the capacity of Mindependent channels and the second is the contribution due tocorrelation. Since the determinant R is always <=1, then correlationalways results in degradation to the ergodic capacity.

An often used channel model for M=2, and 4 called the Kronecker Deltamodel takes this concept further by separating the correlation into twoparts, one near the transmitter and the other near the receiver. Themodel assumes each part to be independent of the other. Two correlationmatrices are defined, one for transmit, R_(T) and one for receiverR_(R). The complete channel correlation is assumed to be equal to theKronecker product of these two smaller matrices.

R _(MIMO) =R _(R) ⊗R _(T)

The correlation among the columns of the H matrix represents thecorrelation between the transmitter and correlation between rows inreceivers. We can write These two one-sided matrices can be written as:

$R_{R =}\frac{1}{\beta}E\left\{ {HH}^{H} \right\}$$R_{T} = {\frac{1}{\alpha}E\left\{ {H^{H}\mspace{14mu} H} \right\}^{T}}$

The constant parameters (the correlation coefficients for each side)satisfy the relationship:

αβ=Tr(R _(MIMO))

Now to see how correlation at the two ends affects the capacity, therandom channel H matrix is multiplied with the two correlation matricesas follows. The matrices can be produced in a number of fashions. Insome cases, test data is available which can be used in the matrix, inothers, a generic form based on Bessel coefficients is used. If thecorrelation coefficient on each side as a parameter is used, eachcorrelation matrix can be written as:

$R_{R} = {{\begin{bmatrix}1 & \sigma & \sigma^{2} \\\sigma & 1 & \sigma \\\sigma^{2} & \sigma & 1\end{bmatrix}\mspace{14mu} {and}\mspace{14mu} R_{R}} = \begin{bmatrix}1 & \beta & \beta^{2} \\\beta & 1 & \beta \\\beta^{2} & \beta & 1\end{bmatrix}}$

Now write the correlated channel matrix in a Cholesky form as

H=√{square root over (R _(T))}H _(W)√{square root over (R _(T))}

where H_(w) is the i.i.d random H matrix, that is now subject tocorrelation effects.

The correlation at the transmitter is mathematically seen as correlationbetween the columns of the H matrix and can written as R_(T). Thecorrelation at the receiver is seen as the correlation between the rowsof the H matrix, R_(R). Clearly if the columns are similar, each antennais seeing a similar channel. When the received amplitudes are similar ateach receiver, correlation at the receiver is seen. The H matrix undercorrelation is ill conditioned, and small changes lead to large changesin the received signal, clearly not a helpful situation.

The capacity of a channel with correlation can be written as:

$C = {\log_{2}\mspace{14mu} \det \mspace{14mu} \left( {1_{NR} + {\frac{SNR}{N_{T}}\mspace{14mu} R_{r}^{1\text{/}2}\mspace{14mu} H^{H}\mspace{14mu} R_{t}^{H\; 1\text{/}2}}} \right)}$

When N_(T)=N_(R) and SNR is high, this expression can be approximatedas:

$C = {{\log_{2}\mspace{14mu} \det \mspace{14mu} \left( {1_{NR} + {\frac{SNR}{N_{T}}\mspace{14mu} H_{u}H_{u}^{H}}} \right)} + {\log_{2}\mspace{14mu} {\det \left( R_{r} \right)}} + {\log_{2}\mspace{14mu} \det \mspace{14mu} \left( R_{t} \right)}}$

The last two terms are always negative since det(R) ≤0. That impliesthat correlation leads to reduction in capacity in frequency selectivechannels as shown in FIG. 157.

Here it is assumed that the frequency response is flat for the durationof the single realization of the H matrix. In FIG. 158 shows a channel15802 that is not flat. Its response is changing with frequency. The Hmatrix now changes within each sub-frequency of the signal. Note thatthis is not time, but frequency. The H matrix is written as a supermatrix of sub-matrices for each frequency.

Assume we can characterize the channel in N frequency sub-bands. The Hmatrix can now be written as a [(N×N_(R)), (N×N_(T))] matrix. A [3×3] Hmatrix is subdivided into N frequency and is written as an [18×18]matrix, with [3×3] matrices on the diagonal.

Spatial Multiplexing and how it Works

Each of the links in a MIMO system is assumed to transmit the sameinformation. This is an implicit assumption of obtaining diversity gain.Multicasting provides diversity gain but no data rate improvement. Ifindependent information could be sent across the antennas, then there isan opportunity to increase the data rate as well as keep some diversitygain. The data rate improvement in a MIMO system is called SpatialMultiplexing Gain (SMG).

The data rate improvement is related to the number of pairs of theRCV/XMT (receive/transmit) antennas, and when these numbers are unequal,it is proportional to smaller of the two numbers, N_(T), N_(R). This iseasy to see; the system can only transmit as many different symbols asthere are transmit antennas. This number is limited by the number ofreceive antennas, if the number of receive antennas is less than thenumber of transmit antennas.

Spatial multiplexing means the ability to transmit higher bit rate whencompared to a system where we only get diversity gains because oftransmissions of the same symbol from each transmitter. Therefore:

$d = {- {\lim_{{SNR}\rightarrow\infty}\mspace{14mu} {\log \frac{{BER}({SNR})}{SNR}\mspace{14mu} {Diversity}\mspace{14mu} {Gain}}}}$$s = {\lim_{{SNR}\rightarrow\infty}{\frac{{Data}\mspace{14mu} {{Rate}({SNR})}}{\log ({SNR})}\mspace{14mu} {Spatial}\mspace{14mu} {Multiplexing}\mspace{14mu} {Gain}}}$

Should the diversity gain or multiplexing gain or maybe a little of bothbe used? The answer is that a little bit of both may be used.

One way to increase the number of independent Eigen channels is to use aset of orthogonal modes. Such a system transmits multiplecoaxially-propagating spatial modes each carrying an independent datastream through a single aperture pair. Therefore, the total capacity ofthe communication system can be increased by a factor equal to thenumber of transmitted modes. An orthogonal spatial modal basis set thathas gained interest recently is orbital angular momentum (OAM). Anelectromagnetic beam with a helical wavefront carries an OAMcorresponding to lℏper photon, where ℏ is the reduced Planck constantand l is an unbounded integer. Importantly, OAM modes with different lvalues are mutually orthogonal, which allows them to be efficiently(de)multiplexed with low inter-modal crosstalk, thereby avoiding the useof multiple-input multiple-output (MIMO) processing.

Another approach for simultaneously transmitting multiple independentdata streams is to use MIMO-based spatial multiplexing, for whichmultiple aperture elements are employed at transmitter/receiver. As awell-established technique in wireless systems, this approach couldprovide capacity gains relative to single aperture systems and increaselink robustness for point-to-point (P2P) communications. In such asystem, each data-carrying beam is received by multiple spatiallyseparated receivers and MIMO signal processing is critical for reducingthe crosstalk among channels and thus allows data recovery.

However, MIMO signal processing becomes more onerous for MIMO-basedspatial multiplexing as the number of aperture elements increases. Inaddition, for OAM multiplexed systems, the detection of high-order OAMmodes presents a challenge for the receiver because OAM beams withlarger l values diverge more during propagation. Therefore, theachievable number of data channels for each type of multiplexingtechnique might be limited, and achieving a larger number of channels byusing any one approach would be significantly more difficult. Similar tothe multiplexing in few-mode and multi-core fibers, these two forms ofspatial multiplexing might be compatible with each other. Thecombination of them by fully exploiting the advantages of eachtechnique, such that they complement each other, might enable a densespatial multiplexed FSO system.

Antenna Placements in MIMO

FIGS. 159 and 160 illustrates the placement of antennas in a MIMOsystem. FIG. 159 illustrates antennas Tx₀ through Tx_(n-1) and receiversRx₀ through Rx_(n-1). FIG. 160 illustrates the transmission pathsbetween transmitters Tx₀ and Tx₁ and receiver Rx.

The vectors describing the antenna placements are given by:

a _(n) ^(t) =nd _(t) sin(θ_(t))n _(x) +nd _(t) cos(θ_(t))n _(z)

a _(m) ^(r) =[D+md _(r) sin(θ_(r))cos(ϕ_(r))]n _(x) +md _(r) cos(θ_(t))n_(z) +md _(r) sin(θ_(r))sin(ϕ_(r))n _(y)

The Euclidean distance between the antennas is:

 d_(nm) = |a_(m)^(r) − a_(n)^(t)| = [(D + md_(r)  sin (θ_(r))  cos (φ_(r)) − nd_(t)  sin (θ_(t)))² + (md_(r)  sin (θ_(r))  sin (φ_(t)))² + (md_(r)  cos (θ_(r)) − nd_(t)  cos (θ_(t)))²]^(1/2)

Since distance D is much larger than the antenna spacing, then:

$\begin{matrix}{d_{nm} \approx {D + {{md}_{r}\sin \; \theta_{r}\cos \; \varphi_{r}} - {{nd}_{t}\sin \; \theta_{t}} + {\frac{1}{2D}\left\lbrack {\left( {{{md}_{r}\cos \; \theta_{r}} - {{nd}_{t}\cos \; \theta_{t}}} \right)^{2} + \left( {{md}_{r}\sin \; \theta_{r}\sin \; \varphi_{r}} \right)^{2}} \right\rbrack}}} \\{{= {D + {{md}_{r}\sin \; \theta_{r}\cos \; \varphi_{r}} - {{nd}_{t}\sin \; \theta_{t}}}}} \\{{+ {\frac{1}{2D}\left\lbrack {m^{2}d_{r}^{2}\mspace{14mu} \cos^{2}\mspace{14mu} \theta_{r}} \right.}}} \\\left. {{{+ n^{2}}d_{t}^{2}\mspace{14mu} \cos^{2}\mspace{14mu} \theta_{t}} - {2{mnd}_{t}d_{r}\cos \; \theta_{t}\cos \; \theta_{r}} + {m^{2}d_{r}^{2}\mspace{14mu} \sin^{2}\mspace{14mu} \theta_{r}\mspace{14mu} \sin^{2}\mspace{14mu} \varphi_{r}}} \right\rbrack\end{matrix}$

Now criteria for the optimal antenna separation can be found. This isachieved by maximizing the capacity as a function of antenna separation.That is to maximize the product of the eigenvalues.

$W = \left\{ \begin{matrix}{{{HH}^{H},}\mspace{14mu}} & {N \leq M} \\{{H^{H}\mspace{14mu} H},} & {N > M}\end{matrix} \right.$

This is obtained if H has orthogonal rows for N≤M or orthogonal columnsfor N >M. Defining the rows of H_(LOS) as h_(n) the orthogonalitybetween them can be expressed as:

$\begin{matrix}{{\text{<}h_{n}},{{h_{i}\text{>}_{n \neq i}} = {\sum\limits_{m = 0}^{M - 1}\; {\exp \left( {{jk}\left( {d_{nm} - d_{im}} \right)} \right)}}}} \\{{= {\sum\limits_{m = 0}^{M - 1}\; {\exp \mspace{14mu} \left( {j\; 2\pi \frac{d_{t}d_{r}\mspace{14mu} {\cos \left( \theta_{r} \right)}\mspace{14mu} {\cos \left( \theta_{t} \right)}}{\lambda \; D}\left( {i - n} \right)m} \right)}}}} \\{{{{\cdot \exp}\mspace{14mu} \left( {{jk}\left\lbrack {{\left( {i - n} \right)d_{t}\mspace{14mu} {\sin \left( \theta_{t} \right)}} + {\frac{1}{2D}\left( {i - n} \right)^{2}d_{t}\mspace{14mu} {\cos^{2}\left( \theta_{t} \right)}}} \right\rbrack} \right)} = 0}} \\{{\left. \Rightarrow\frac{\sin \left( {{kd}_{t}d_{r}\mspace{14mu} {\cos \left( \theta_{r} \right)}\mspace{14mu} {\cos \left( \theta_{t} \right)}\mspace{14mu} \left( {i - n} \right)M\text{/}2D} \right)}{\sin\left( {{kd}_{t}d_{r}\mspace{14mu} {\cos \left( \theta_{r} \right)}\mspace{14mu} {\cos \left( \theta_{t} \right)}\mspace{14mu} \left( {i - n} \right)\text{/}2D} \right.} \right. = {\left. 0\Rightarrow{d_{t}d_{r}} \right. = {\frac{\lambda \; D}{M\; {\cos \left( \theta_{t} \right)}{\cos \left( \theta_{r} \right)}}K}}}}\end{matrix}$

where K is a positive odd number usually chosen to be 1 since that givesthe smallest optimal antenna separation. Doing a similar derivation forthe case N >M would give the same expression only with M instead of N.Defining V=min(M,N), the general expression for antenna separation is:

${d_{t}d_{r}} = {\frac{\lambda \; D}{V\; {\cos \left( \theta_{t} \right)}{\cos \left( \theta_{r} \right)}}K}$

That is the separation increases by distance D and decreases asfrequency increases.

Defining a new parameter η as:

$\eta = \sqrt{\frac{d_{t}d_{r}}{\left( {d_{t}d_{r}} \right)_{opt}}}$

To quantify the deviation from optimality. Choosing d_(t)=d_(r)=dThen this reduces to η=d/d_(opt)Then the condition number for 2×2 MIMO will be

$\kappa = \sqrt{\frac{2 + \left\lbrack {2 + {2\mspace{14mu} {\cos \left( {\pi\eta}^{2} \right)}}} \right\rbrack^{1\text{/}2}}{2 - \left\lbrack {2 + {2\mspace{14mu} {\cos \left( {\pi\eta}^{2} \right)}}} \right\rbrack^{1\text{/}2}}}$

Calculating the distance d₁ and d₂ we have:

d ₁ ² =d _(t) ²/4+D ² −d _(t) D cos(ϕ)

d ₂ ² =d _(t) ²/4+D ² +d _(t) D cos(ϕ)

The gain relative to single RX antenna can be expressed as:

${G\left( {\varphi,\alpha} \right)} = {{\frac{1}{T_{sc}}{\int_{0}^{T_{sc}}{\left( {{\sin \left( {{\omega \; t} + {d_{1}k}} \right)} + {\sin \left( {{\omega \; t} + {d_{2}k} + \alpha} \right)}} \right)^{2}{dt}}}} = {{{\cos \left( {{d_{1}k} - {d_{2}k} - \alpha} \right)} + 1} = {{\cos \left( {{k\; \Delta \; d} + \alpha} \right)} + 1}}}$

Where:

Δd=d ₂ −d ₁.

But:

$\begin{matrix}{\left( {d_{2} - d_{1}} \right)^{2} = {\left( {d_{1}^{2} + d_{2}^{2} - {2d_{1}d_{2}}} \right) = {\frac{d_{t}^{2}}{2} + {2D^{2}} - \sqrt{\frac{d_{t}^{4}}{4} + {4D^{4}} + {2d_{t}^{2}D^{2}} - {4d_{t}^{2}D^{2}\mspace{14mu} {\cos^{2}(\varphi)}}}}}} \\{{{\approx {\frac{d_{t}^{2}}{2} + {2D^{2}} - {2D^{2}} - \frac{{d_{t}^{4}\text{/}4} + {2d_{t}^{2}D^{2}} - {4d_{t}^{2}D^{2}\mspace{14mu} {\cos^{2}(\varphi)}}}{2D^{2}}}} = {{d_{t}^{2}\mspace{14mu} {\cos^{2}(\varphi)}} - \frac{d_{t}^{4}}{16D^{2}}}}}\end{matrix}$

For D» d_(t) ²:

Δd≈d _(t) cos(ϕ)

Therefore:

${G\left( {\varphi,\alpha} \right)} = {{{\cos \left( {{k\; \Delta \; d} + \alpha} \right)} + 1} \approx {{\cos \mspace{14mu} \left( {{\frac{2\pi}{\lambda}\mspace{14mu} d_{t}\mspace{14mu} {\cos (\varphi)}} + \alpha} \right)} + 1}}$

Wideband MIMO Model

For a narrowband MIMO we had r=sH+n which can be expressed as:

${r_{m}\lbrack j\rbrack} = {\sum\limits_{n = 1}^{N}\; {h_{nm}{s_{n}\lbrack j\rbrack}}}$

Where j is the discrete symbol timing.

Now a wideband channel will act as a filter so that the gains need bereplaced by filters as:

${r_{m}\lbrack j\rbrack} = {\left. {\sum\limits_{n = 1}^{N}\; {\sum\limits_{i = {- L_{t}}}^{L_{t}}\; {{h_{nm}\left\lbrack {i,j} \right\rbrack}{s_{n}\left\lbrack {j - i} \right\rbrack}}}}\Leftrightarrow r_{m} \right. = {\sum\limits_{n = 1}^{N}\; {h_{nm}*s_{n}}}}$

Transmitting L_(B) symbols (block length) the channel matrix can beexpressed as:

$\Psi_{nm} = \begin{bmatrix}{h_{nm}\lbrack 0\rbrack} & \ldots & {h_{nm}\left\lbrack {- L_{t}} \right\rbrack} & 0 & \ldots & 0 \\{h_{nm}\lbrack 1\rbrack} & {h_{nm}\lbrack 0\rbrack} & \ldots & {h_{nm}\left\lbrack {- L_{t}} \right\rbrack} & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & {h_{nm}\left\lbrack L_{t} \right\rbrack} & \ldots & {h_{nm}\lbrack 0\rbrack}\end{bmatrix}^{T}$ ${Or} = \begin{bmatrix}\Psi_{11} & \ldots & \Psi_{1M} \\\vdots & \ddots & \vdots \\\Psi_{N\; 1} & \ldots & \Psi_{NM}\end{bmatrix}$

Defining signal matrix as:

s=[s ₁ , . . . , s _(N)]

s _(n) =[s _(n)[1], . . . ,s _(n) [L _(B)]]

Then the noise free received signal can be expressed as:

r=sΨ

For frequency selective channel, the capacity becomes frequencydependent. For a SISO channel, the capacity becomes:

C=∫ _(−∞) ^(∞) log₂(1+γ|H(f)|²)df

And therefore for a MIMO channel the capacity becomes

$C = {\sum\limits_{i = 1}^{\min {({N,M})}}\; {\int_{- \infty}^{\infty}{{\log_{2}\left( {1 + {{\gamma\lambda}_{i}(f)}} \right)}{df}}}}$

Line of Sight MIMO Channel

In LOS-MIMO there is one dominating path so that the incoming phases aredominated by the geometry of the channel and not scattering. Other thanthe LOS-path, there can be secondary paths caused by atmosphericscintillation or a reflection off the ground. This results in afrequency selective channel. The common model to use is a plane earthmodel with one reflection from ground. The two paths have some delaydifference τ which is typically set to 6.3 nsec in Rummler's model. Theimpulse response of such a channel is:

h(t)=δ(t)+bδ(t−τ)e ^(jϕ) ⇔H(s)=L{h(t)}=1+be ^(−τs+jϕ)

that is the channel introduces periodic notches in frequency spectrum.If one of the notches happen to be within the bandwidth, the channelwill suffer from sever fading and ISI.

Dual Polarized MIMO

For a 4×4 Dual polarized MIMO, one can achieve four fold capacitycompared to SISO. The narrowband channel matrix can be written as:

$H = {\begin{bmatrix}h_{{1V},{1V}} & h_{{1V},{1H}} & h_{{1V},{2V}} & h_{{1V},{2H}} \\h_{{1H},{1V}} & h_{{1H},{1H}} & h_{{1H},{2V}} & h_{{1H},{2H}} \\h_{{2V},{1V}} & h_{{2V},{1H}} & h_{{2V},{2V}} & h_{{2V},{2H}} \\h_{{2H},{1V}} & h_{{2H},{1H}} & h_{{2H},{2V}} & h_{{2H},{2H}}\end{bmatrix} = {\begin{bmatrix}H_{11} & H_{12} \\H_{21} & H_{22}\end{bmatrix} = {\quad\begin{bmatrix}{\sqrt{1 - \alpha}e^{{jkd}_{11}}} & {\sqrt{\alpha}e^{{jkd}_{11}}} & {\sqrt{1 - \alpha}e^{{jkd}_{12}}} & {\sqrt{\alpha}e^{{jkd}_{12}}} \\{\alpha \; e^{{jkd}_{11}}} & {\sqrt{1 - \alpha}e^{{jkd}_{11}}} & {\sqrt{\alpha}e^{{jkd}_{12}}} & {\sqrt{1 - \alpha}e^{{jkd}_{12}}} \\{\sqrt{1 - \alpha}e^{{jkd}_{21}}} & {\sqrt{\alpha}\overset{{jkd}_{21}}{e}} & {\sqrt{1 - \alpha}e^{{jkd}_{22}}} & {\sqrt{\alpha}e^{{jkd}_{22}}} \\{\alpha \; e^{{jkd}_{21}}} & {\sqrt{1 - \alpha}e^{{jkd}_{21}}} & {\sqrt{\alpha}e^{{jkd}_{22}}} & {\sqrt{1 - \alpha}e^{{jkd}_{22}}}\end{bmatrix}}}}$

Where:

$H = {{H_{LoS} \otimes W_{XPD}} = {\begin{bmatrix}e^{{jkd}_{11}} & e^{{jkd}_{12}} \\e^{{jkd}_{21}} & e^{{jkd}_{22}}\end{bmatrix} \otimes \begin{bmatrix}\sqrt{1 - \alpha} & \sqrt{\alpha} \\\sqrt{\alpha} & \sqrt{1 - \alpha}\end{bmatrix}}}$

Here α measures the ratio of the power for one polarization that istransferred to the other polarization. Then:

$\alpha = \frac{1}{{XPD} + 1}$

And condition numbers can be calculated as:

c ₁=(2−f _(a)(a))[1−cos(πη²/2)]

c ₂=(2−f _(a)(a))[cos(πη²/2)+1]

c ₃=(2+f _(a)(a))[cos(πη²/2)+1]

c ₄=(2+f _(a)(a))[1−cos(πη²/2)]

Where

$\kappa = \sqrt{\frac{\max \left( {c_{1},c_{2},c_{3},c_{4}} \right)}{\min \left( {c_{1},c_{2},c_{3},c_{4}} \right)}}$

For SISO, assuming Gray coding and square QAM-M, the BER can becalculated as

$P_{b} = {\left( {1 - \frac{1}{\sqrt{M}}} \right){Q\left( \sqrt{\frac{3k}{M - 1}\frac{E_{b}}{N_{0}}} \right)}}$

Therefore, the BER for a MIMO system using SVD can be expressed as:

$P_{b} = {\frac{1}{R_{H}}{\sum\limits_{i = 1}^{R_{H}}{\left( {1 - \frac{1}{\sqrt{M}}} \right){Q\left( \sqrt{\frac{3k}{M - 1}\frac{E_{b}}{N_{0}}\sigma_{i}^{2}} \right)}}}}$

Referring now to FIG. 161 by combining mode division multiplexingtechniques one 6102 such as those described previously for combiningmultiple input signals onto a single carrier signal by applying adifferent orthogonal functions, such as Hermite Gaussian functions andLaguerre Gaussian functions, to each of the multiple input signals, withmultiple input multiple output (MIMO) techniques 16102 increasedbandwidth 16106 may be achieved for free space communications. Thus, asillustrated in FIG. 162, multiple input signals 16202 are applied to MDMprocessing circuitry 16204. The MDM processing circuitry 16204 usesHermite Gaussian functions, Laguerre Gaussian functions or otherorthogonal functions to generate an HG, LG or orthogonal transmissionbeam. The MDM processed beam from the MDM processing circuit 16204 isprovided to a MIMO transmitter 16206 for transmission of an outputsignal. The combined MIMO+MDM multiplexing enhances the performance offree space point to point communication systems by fully exploiting theadvantages of each multiplexing technique.

The transmission 16208 from the MIMO transmitter 16206 to a MIMOreceiver 16210 can be done at both RF as well as optical frequencies.The MIMO transmitter 16206 and the MIMO receiver 16210 each includemultiple antennas associated with each of the transmission channels.There is an optimal way to place each element of the antenna array onboth the transmitter and receiver sides to maximize capacity andminimize correlations. Also, game theoretical algorithms such as waterfilling algorithms can be used to find the optimum distribution of powerin each of the radio paths in the channel matrix between the transmitterand the receiver. The received signals are separated as discussedhereinabove at the MIMO receiver 16210 and the received signals are thenmultiplexed using MDM processing circuitry 16212 at the receiver side.This ultimately receives the originally received multiple input signalsas multiple output signals 16214. Interchannel crosstalk's effects areminimized by the OAM beams inherent orthogonality provided by the MDMprocessing circuitry 16204 and by the MIMO signal processing. The OAMand MIMO-based spatial multiplexing can be compatible with andcomplement each other thereby providing for a dense or super massivecompactified MIMO system.

Further improvements in signal to noise ratio and a creation of adenser, super compactified MIMO system may be achieved, as illustratedin FIG. 163. Multiple input signals 16302 are applied to MDM processingcircuitry 16304. The MDM processing circuitry 16204 uses HermiteGaussian functions, Laguerre Gaussian functions or other orthogonalfunctions to generate an HG, LG or orthogonal transmission beam. The MDMprocessed beam from the MDM processing circuit 16304 is provided to amaximum ratio combining circuit that processes the signal as describedin the attached appendix. The MRC processed signal is provided to a MIMOtransmitter 16306 for transmission of an output signal. The combinedMIMO+MRC+MDM multiplexing enhances the performance of free space pointto point communication systems by fully exploiting the advantages ofeach multiplexing technique and creates a super dense data transmissionsignal.

The transmission 16308 from the MIMO transmitter 16306 to a MIMOreceiver 16310 can be done at both RF as well as optical frequencies.The received signals are processed by an MRC circuit 16311 and separatedas discussed hereinabove at the MIMO receiver 16310. The receivedsignals are then multiplexed using MDM processing circuitry 16212 at thereceiver side. This ultimately receives the originally received multipleinput signals as multiple output signals 16214. Interchannel crosstalk'seffects are minimized by the beams inherent orthogonality provided bythe MDM processing circuitry 16204 and by the MIMO signal processing.The Orthogonal signal, MRC and MIMO-based spatial multiplexing can becompatible with and complement each other thereby providing for a denseor super massive compactified MIMO system.

The MIMO+MDM+MRC combination can work in both time, frequency, spaceangular and polarization diversity with beamforming and null stearing.The combination of MIMO+MDM+MRC can also be configured to provide arraygain based on the configuration of the transmit and receive arrays. Thecombined MIMO+MDM+MRC channel can be decomposed into several independentparallel channels each of which can be thought of as an SISO channel.Singular value decomposition (SVD) techniques can be used to decomposedthe combined MIMO+MDM+MRC channel matrix. The combined MIMO+MDM+MRCchannel requires the use of pre-coding at the transmitter 16306 andreceiver shaping at the receiver 16310. MMSE techniques can be used forcombined MIMO+MDM+MRC systems which can be like a CF receiver at highSNR and an MRC receiver at low SNR. Additionally, a space timeAlamuti-type technique can be applied to the combined MIMO+MDM+MRCchannel. Existing manufacturer chipsets for example, Qualcomm orBroadcom, can be used for all capacity enhancements using MIMO. However,the SNR of each input to the chipsets may be increased using RF frontend that performs MRC by considering additional antennas at each port.One can further perform carrier aggregation using carrier aggregationtechniques in LTE-advanced or chipsets such as Qualcomm's multi-firechipset that perform carrier aggregation without and LTE anchor channel.This can prove to be beneficial when combined with MIMO+MDM+MRCtechniques or any combination of techniques.

Referring now also to FIG. 164, the multiple input signals 16402 areapplied to MRC processing circuitry 16404. The MRC processing circuitry16404 processes the signals as discussed in the appendix. The MRCprocessing circuitry improves the signal to noise ratio. The MRCprocessed beam from the MRC processing circuit 16404 is provided to amaximum ratio combining circuit that processes the signal as describedin the attached appendix. The MRC processed signal is provided to a MIMOtransmitter 16406 for transmission of an output signal. The combinedMIMO+MRC enhances the performance of free space point to pointcommunication systems by fully exploiting the advantages of eachmultiplexing technique and creates a super dense data transmissionsignal.

The transmission 16408 from the MIMO transmitter 16406 to a MIMOreceiver 16410 can be done at both RF as well as optical frequencies.The received signals are processed by an MRC circuit 16411. Thisultimately provides the originally received multiple input signals asmultiple output signals 16414. The MRC and MIMO-based spatialmultiplexing can be compatible with and complement each other therebyproviding for a dense or super massive compactified MIMO system.

It will be appreciated by those skilled in the art having the benefit ofthis disclosure that this system and method for communication usingorbital angular momentum with multiple layer overlay modulation providesimproved bandwidth and data transmission capability. It should beunderstood that the drawings and detailed description herein are to beregarded in an illustrative rather than a restrictive manner, and arenot intended to be limiting to the particular forms and examplesdisclosed. On the contrary, included are any further modifications,changes, rearrangements, substitutions, alternatives, design choices,and embodiments apparent to those of ordinary skill in the art, withoutdeparting from the spirit and scope hereof, as defined by the followingclaims. Thus, it is intended that the following claims be interpreted toembrace all such further modifications, changes, rearrangements,substitutions, alternatives, design choices, and embodiments.

APPENDIX

$\begin{bmatrix}y_{1} \\y_{2} \\\vdots \\y_{r}\end{bmatrix} = {{\begin{bmatrix}h_{11} & h_{12} & \; & \ldots & h_{1t} \\h_{21} & h_{22} & \; & \; & \; \\h_{31} & \; & \ddots & \; & \; \\\vdots & \; & \; & \ddots & \; \\h_{r\; 1} & \; & \; & \; & h_{rt}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2} \\\vdots \\x_{t}\end{bmatrix}} + \begin{bmatrix}n_{1} \\n_{2} \\\vdots \\n_{r}\end{bmatrix}}$ receive  channel  matrix  transmit  noise$\overset{\rightarrow}{y} = {{H\overset{\rightarrow}{x}} + \overset{\rightarrow}{n}}$

Spatially uncorrelated noise (Isotropic—equally distributed uniformly inall directions)

$R_{n} = {{E\left( {nn}^{H} \right)} = {{\overset{\rightarrow}{n}{\overset{\rightarrow}{n}}^{H}} = {\begin{bmatrix}\sigma_{n^{2}} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \sigma_{n^{2}}\end{bmatrix} = {{\sigma_{n^{2}}\begin{bmatrix}1 & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & 1\end{bmatrix}} = {\sigma_{n^{2}}1}}}}}$

MIMO Receiver: Linear Receiver

H ⁻¹ [{right arrow over (y)}]=H ⁻¹ H{right arrow over (x)}+H ⁻¹ {rightarrow over (n)}

H ⁻¹ {right arrow over (y)}={right arrow over (x)}+H ⁻¹ {right arrowover (n)}

1. Inverse only exists for square matrices r=t2. Inverse only exists if the matrix has full rank (independent rows)

To define a generalized inverse r ≥t

$\begin{bmatrix}y_{1} \\y_{2} \\\vdots \\y_{r}\end{bmatrix} = {{\begin{bmatrix}h_{11} & \ldots & h_{1t} \\\vdots & \ddots & \; \\h_{r\; 1} & \; & \ddots\end{bmatrix}\begin{bmatrix}x_{1} \\\vdots \\x_{t}\end{bmatrix}} + \overset{\rightarrow}{n}}$

Thin matrix:

$\quad\begin{bmatrix}\; & \; & \; \\\; & \; & \; \\\; & \; & \; \\\; & \; & \;\end{bmatrix}$

more rows than columns, more equations than unknownWhen r ≥t there are more equations than unknownsequations are r −measurementsunknown are t −transmittershence, there might not be an exact solution and therefore the solutionscan approximate with some errorAmongst all possible tx vectors {right arrow over (x)}, choose theminimum error vector

error=∥{right arrow over (y)}−H{right arrow over (x)}∥ ² least-squaressolution

measurement unknown vector

choose an {right arrow over (x)} that minimizes the norm

Vector Differentiation

${Given}\mspace{14mu} {f\left( \overset{\rightarrow}{x} \right)}$$\frac{{df}\left( \overset{\rightarrow}{x} \right)}{d\overset{\rightarrow}{x}} = {\overset{\rightarrow}{x} = {\left. \begin{bmatrix}x_{1} \\x_{2} \\\vdots \\x_{t}\end{bmatrix}\Rightarrow\frac{df}{d\overset{\rightarrow}{x}} \right. = \begin{bmatrix}\frac{df}{{dx}_{1}} \\\vdots \\\frac{df}{{dx}_{t}}\end{bmatrix}}}$${f\left( \overset{\rightarrow}{x} \right)} = {{c^{T}\left( \overset{\rightarrow}{x} \right)} = {{{c_{1}x_{1}} + {c_{2}x_{2}} + {\ldots \mspace{14mu} c_{t}x_{t}}} = {{{\overset{\rightarrow}{x}}^{T}\overset{\rightarrow}{c}} = {{\overset{\rightarrow}{c}}^{T}\overset{\rightarrow}{x}}}}}$$\frac{d\left( {c^{T}\overset{\rightarrow}{x}} \right)}{d\overset{\rightarrow}{x}} = {\frac{d\left( {{\overset{\rightarrow}{x}}^{T}\overset{\rightarrow}{c}} \right)}{d\overset{\rightarrow}{x}} = {\begin{bmatrix}c_{1} \\c_{2} \\\vdots \\c_{t}\end{bmatrix} = \overset{\rightarrow}{c}}}$$\frac{d\left( {{\overset{\rightarrow}{x}}^{H}\overset{\rightarrow}{c}} \right)}{d{\overset{\rightarrow}{x}}^{H}} = \overset{\rightarrow}{c}$

minimize ∥{right arrow over (y)}−H{right arrow over (x)}∥²For real matrices

$\begin{matrix}{{{\overset{\rightarrow}{y} - {H\overset{\rightarrow}{x}}}}^{2} = {\left( {\overset{\rightarrow}{y} - {H\overset{\rightarrow}{x}}} \right)^{T}\left( {\overset{\rightarrow}{y} - {H\overset{\rightarrow}{x}}} \right)}} \\{= {{{\overset{\rightarrow}{y}}^{T}\overset{\rightarrow}{y}} - {{\overset{\rightarrow}{y}}^{T}H\overset{\rightarrow}{x}} - {{\overset{\rightarrow}{x}}^{T}H^{T}\overset{\rightarrow}{y}} + {{\overset{\rightarrow}{x}}^{T}H^{T}H\overset{\rightarrow}{x}}}}\end{matrix}$

differentiate this with respect to {right arrow over (x)} and set thederivative equal to zero

$\frac{d\left( {{\overset{\rightarrow}{y} - {H\overset{\rightarrow}{x}}}}^{2} \right)}{d\overset{\rightarrow}{x}} = {{{{- 2}H^{T}\overset{\rightarrow}{y}} + {2H^{T}H\overset{\rightarrow}{x}}} = 0}$${\left( {H^{T}H} \right)\overset{\rightarrow}{x}} = {H^{T}\overset{\rightarrow}{y}}${right arrow over (x)}=(H ^(T) H)⁻¹ H ^(T) {right arrow over (y)}approximate solution that minimizes error

-   -   zero-forcing RX        For complex channel matrices, use the Tranpose Conjugate

{right arrow over (x)}=(H ^(H) H)⁻¹ H ^(H) {right arrow over (y)}pseudo-inverseH ^(†)

H ^(†)=(H ^(H) H)⁻¹ H ^(H)(pseudo-inverse)(left-inverse)H ^(†) y=x

H ^(†) H=(H ^(H) H)⁻¹ H ^(H) H=1 if H ⁻¹ does not exist then this ismore general inverse (pseudo inverse).

B=A ⁻¹

BA=1

AB=1

Inverse is unique, however the pseudo-inverse is not.If H⁻¹ exists then (H^(H)H)⁻¹H^(H)=H⁻¹H^(−H)·H^(H)=H⁻¹Then pseudo-inverse=inverseDiversity order of ZF-receiver

diversity=r−t+1 if r=4t=2

diversity=4−2+1=3

if r=t diversity=1 low because of noise amplificationFor SISO y=hx+n if h=small, this goes to infinity

${h^{- 1}y} = {\frac{x}{h} + \frac{n}{h}}$$\frac{n}{h}->{{noise}\mspace{14mu} {amplification}}$

Disadvantages of ZF receiver is that it results in noise amplification.Therefore, look for a better receiver algorithm for MIMO called MinimumMean Square Error (MMSE) ReceiverEstimate x given

$\quad\begin{bmatrix}y_{1} \\y_{2} \\\vdots \\y_{r}\end{bmatrix}$

r-measurements to find {right arrow over (x)}

Let's find a Linear Estimator to estimate {right arrow over (x)}

c ^(T) {right arrow over (y)}

choose {right arrow over (c)} such so that

E{x̂ − x²}minimum$E{\left\{ {{{c^{T}\overset{\rightarrow}{y}} - \overset{\rightarrow}{x}}}^{2} \right\} {minimum}}$$\begin{matrix}{{E\left\{ {{{c^{T}\overset{\rightarrow}{y}} - \overset{\rightarrow}{x}}}^{2} \right\}} = {E\left\{ {\left( {{c^{T}\overset{\rightarrow}{y}} - \overset{\rightarrow}{x}} \right)^{T}\left( {{c^{T}\overset{\rightarrow}{y}} - \overset{\rightarrow}{x}} \right)} \right\}}} \\{= {E\left\{ {{{\overset{\rightarrow}{c}}^{T}\overset{\rightarrow}{y}{\overset{\rightarrow}{y}}^{T}\overset{\rightarrow}{c}} - {\overset{\rightarrow}{x}{\overset{\rightarrow}{y}}^{T}\overset{\rightarrow}{c}} - {{\overset{\rightarrow}{c}}^{T}\overset{\rightarrow}{y}{\overset{\rightarrow}{x}}^{T}} + {\overset{\rightarrow}{x}{\overset{\rightarrow}{x}}^{T}}} \right\}}}\end{matrix}$Define: E({right arrow over (y)}{right arrow over (y)} ^(H))=R _(yy)

E({right arrow over (x)}{right arrow over (y)} ^(H))=R _(xy)

E({right arrow over (y)}{right arrow over (x)} ^(H))=R _(yx) =R _(xy)^(T)

E({right arrow over (x)}{right arrow over (x)} ^(H))=R _(xx)

$\begin{matrix}{{E\left\{ {{\hat{x} - x}}^{2} \right\}} = {{c^{T}R_{yy}c} - {R_{xy}c} - {c^{T}R_{yx}} + {R_{xx}{minimize}}}} \\{= {{c^{T}R_{yy}c} - {2\; c^{T}R_{yx}} + {{R_{xx}{minimize}}\mspace{14mu} {f(c)}}}}\end{matrix}$

Take derivative with respect to {right arrow over (c)} and set to zero.

=2R _(yy) c−2R _(yx)=0

R _(yy) c=R _(yx) {right arrow over (c)}=R _(yy) ⁻¹ R _(yx) linear MMSEestimator

{tilde over ({right arrow over (x)})}=c ^(T) {right arrow over (y)}

{tilde over ({right arrow over (x)})}=c ^(H) {right arrow over (y)} forcomplex (more general)

{tilde over ({right arrow over (x)})}=(R _(yy) ⁻¹ R _(yx))^(H) {rightarrow over (y)}=R _(xy) R _(yy) ⁻¹ {right arrow over (y)}

{right arrow over (y)}=H{right arrow over (x)}+{right arrow over (n)}TX

E(xx ^(H))=E({right arrow over (x)}{right arrow over (x)}^(H))=covariance

${E\left( {xx}^{H} \right)} = {{E\left( {\overset{\rightarrow}{x}{\overset{\rightarrow}{x}}^{H}} \right)} = {{covariance} = {{E\left\{ {\begin{bmatrix}x_{1} \\x_{2} \\\vdots \\x_{t}\end{bmatrix}\left\lbrack {x_{1}^{*}\mspace{14mu} x_{2}^{*}\mspace{14mu} \ldots \mspace{14mu} x_{t}^{*}} \right\rbrack} \right\}} = {R_{xx} = {{E\left\{ \begin{bmatrix}{x_{1}}^{2} & {x_{1}x_{2}^{*}} & \ldots \\{x_{2}x_{1}^{*}} & {x_{2}}^{2} & \; \\\vdots & \; & {x_{t}}^{2}\end{bmatrix} \right\}} = {\begin{bmatrix}P_{d} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & P_{d}\end{bmatrix} = {P_{d}1}}}}}}}$

Correlation between symbols x₂x₁* . . . →all zero.

|x ₁|²=tx power=P _(d)

Tx cov=[R _(xx) ]=P _(d)1_(t)

$\begin{matrix}{R_{yy} = {{E\left\{ {yy}^{H} \right\}} = {E\left\{ {\left( {{H\; \overset{\rightarrow}{x}} + \overset{\rightarrow}{n}} \right)\left( {{H\; \overset{\rightarrow}{x}} + \overset{\rightarrow}{n}} \right)^{H}} \right\}}}} \\{= {E\left\{ {{H\; \overset{\rightarrow}{x}{\overset{\rightarrow}{x}}^{H}H^{H}} + {\overset{\rightarrow}{n}{\overset{\rightarrow}{x}}^{H}H^{H}} + {H\; \overset{\rightarrow}{x}{\overset{\rightarrow}{n}}^{H}} + {\overset{\rightarrow}{n}{\overset{\rightarrow}{n}}^{H}}} \right\}}} \\{{{{{no}\mspace{14mu} {correlation}\mspace{14mu} {between}\mspace{14mu} {noise}}\&}\mspace{11mu} {transit}}} \\{{{\overset{\rightarrow}{n}{\overset{\rightarrow}{x}}^{H}} = {{0\mspace{14mu} {and}\mspace{14mu} \overset{\rightarrow}{x}{\overset{\rightarrow}{n}}^{H}} = 0}}} \\{= {{{HR}_{xx}H^{H}} + {\sigma_{n}^{2}1}}}\end{matrix}$R _(yy) =P _(d) HH ^(H)+σ_(n) ²1 covariance matrix of received symbolvectors

$\overset{\rightarrow}{\hat{x}} = {c^{H}\overset{\rightarrow}{y}}$$\overset{\rightarrow}{\hat{x}} = {R_{xy}R_{yy}^{- 1}\overset{\rightarrow}{y}}$$\begin{matrix}{R_{yx} = {E\left\{ {\overset{\rightarrow}{y}{\overset{\rightarrow}{x}}^{H}} \right\}}} \\{= {E\left\{ {\left( {{H\overset{\rightarrow}{x}} + \overset{\rightarrow}{n}} \right){\overset{\rightarrow}{x}}^{H}} \right\}}} \\{= {E\left\{ {{H\overset{\rightarrow}{x}{\overset{\rightarrow}{x}}^{H}} + {\overset{\rightarrow}{n}{\overset{\rightarrow}{x}}^{H}}} \right\}}} \\{= {{HR}_{xx} = {H\left\lbrack {P_{d}1} \right\rbrack}}} \\{= {P_{d}H}}\end{matrix}$$\overset{\rightarrow}{c} = {{R_{yy}^{- 1}R_{yx}} = {\left( {{P_{d}{HH}^{H}} + {\sigma_{n}^{2}1}} \right)^{- 1}\left( {P_{d}H} \right)}}$$\overset{\rightarrow}{c} = {{P_{d}\left( {{HH}^{H} + {\sigma^{2}1}} \right)}^{- 1}H\mspace{14mu} \left( {{MMSE}\mspace{14mu} {estimator}} \right)}$$\overset{\rightarrow}{\hat{x}} = {{\overset{\rightarrow}{c}}^{H}\overset{\rightarrow}{y}}${right arrow over (x)}=P _(d) H ^(H)(HH ^(H)+σ²1)⁻¹ {right arrow over(y)}(linear minimum mean square estimator for MIMO)

H ^(H)(P _(d) HH ^(H)+σ_(n) ²1)⁻¹=(P _(d) H ^(H) H+σ _(n) ²1)⁻¹ H^(H)→(easier to invert or work with)

(P _(d) H ^(H) H+σ _(n) ²1)H ^(H) =H ^(H)(P _(d) HH ^(H)+σ_(n) ²1)

P _(d) H ^(H) HH ^(H)+σ_(n) ² H ^(H) =P _(d) H ^(H) HH ^(H)+σ_(n) ² H^(H)

then

{tilde over ({right arrow over (x)})}=P _(d)(P _(d) H ^(H) H+σ _(n)²1)⁻¹ H ^(H) {right arrow over (y)}(linear minimum mean square estimatorfor MIMO)

Consider H=h for a SISO

$\hat{x} = {{P_{d}\left( \frac{h^{*}}{{P_{d}{h}^{2}} + \sigma_{n}^{2}} \right)}\overset{\rightarrow}{y}}$$\hat{x} = {\frac{P_{d}h^{*}}{\sigma_{n}^{2}}\overset{\rightarrow}{y}\mspace{14mu} \left( {{{{if}\mspace{14mu} h}{small}},{{this}\mspace{14mu} {does}\mspace{14mu} {not}\mspace{14mu} {blow}\mspace{14mu} {up}}} \right)}$

MIMO MMSE is more general and does not result in noise amplificationwhen h →small

{circumflex over (x)}=P _(d)(P _(d) H ^(H) H+σ _(n) ²1)⁻¹ H ^(H) {rightarrow over (y)}

at high SNR P_(d)=largethen

{tilde over ({right arrow over (x)})}≈P _(d)(P _(d) H ^(H) H)⁻¹ H ^(H)_({right arrow over (y)})≈(H ^(H) H)⁻¹ H ^(H)_({right arrow over (y)})→ZF receiver

at low SNR P_(d)=small

$\begin{matrix}{\overset{\rightarrow}{\hat{x}} = {{P_{d}\left( {{P_{d}H^{H}H} + {\sigma_{n}^{2}1}} \right)}^{- 1}H^{H}\overset{\rightarrow}{y}}} \\{= {{P_{d}\left( {\sigma_{n}^{2}1} \right)}^{- 1}H^{H}\overset{\rightarrow}{y}}} \\{= \left. {\frac{P_{d}}{\sigma_{n}^{2}}H^{H}\overset{\rightarrow}{y}}\mspace{14mu}\rightarrow\mspace{14mu} {{maximum}\mspace{14mu} {ratio}\mspace{14mu} {combining}} \right.} \\{{({MRC})\mspace{14mu} {receiver}\mspace{14mu} {matched}\mspace{14mu} {filter}\mspace{14mu} {results}\mspace{14mu} {in}}} \\{{{maximizing}\mspace{14mu} {signal}\mspace{14mu} {to}\mspace{14mu} {noise}\mspace{14mu} {{ratio}.}}}\end{matrix}\mspace{20mu}$

Decomposition of MIMO Channel H Singular Value Decomposition (SVD)

 = U Σ V^(H) ${U\; \Sigma \; V^{H}} = {{\begin{bmatrix}u_{1} & u_{2} & \ldots & u_{t}\end{bmatrix}\begin{bmatrix}\sigma_{1} & \; & \; \\\; & \ddots & \; \\\; & \; & \sigma_{t}\end{bmatrix}}\begin{bmatrix}v_{1}^{H} \\v_{2}^{H} \\\vdots \\v_{t}^{H}\end{bmatrix}}$∥u _(i)∥²=1 orthonormal u _(i) ^(H) u _(j)=0 if i≠j

∥v _(i)∥²=1 orthonormal v _(i) ^(H)=0 if i≠j

v ^(H) v=vv ^(H)=1 unitary matrix

u ^(H) u=1uu ^(H)≠in generaluu ^(H)=1 only ifr=t generally r>t

Σ is a diagonal matrix

σ₁, σ₂, σ₃ . . . σ_(t) are all singular values of

σ₁>σ₂>σ₃ . . . σ_(t)≥0 all positive and ordered

Eigen value decomposition only works for square matrices, but SVD worksfor matrices of all non-square sizes.Number of singular values=rank of the matrix

SVD

H=UΣV ^(H)

{right arrow over (y)}=H{right arrow over (x)}+{right arrow over (n)}

{right arrow over (y)}=UΣV ^(H) {right arrow over (x)}+{right arrow over(n)}

at receiver, multiply {right arrow over (y)} by U^(H)

U ^(H) {right arrow over (y)}=U ^(H) UΣV ^(H) {right arrow over (x)}+U^(H) {right arrow over (n)}

{tilde over ({right arrow over (y)})}=ΣV ^(H) {right arrow over (x)}+U^(H) {right arrow over (n)}

Now, let's do some pre-coding at TX

{right arrow over (x)}=V{tilde over ({right arrow over (x)})}

{tilde over ({right arrow over (y)})}=ΣV ^(H) V{tilde over ({right arrowover (x)})}+{tilde over ({right arrow over (n)})}

{tilde over ({right arrow over (y)})}=Σ{tilde over ({right arrow over(x)})}+{tilde over ({right arrow over (n)})}

$\begin{bmatrix} \\ \\\vdots \\

\end{bmatrix} = {{\begin{bmatrix}\sigma_{1} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \sigma_{t}\end{bmatrix}\begin{bmatrix} \\ \\\vdots \\

\end{bmatrix}} + \begin{bmatrix} \\ \\\vdots \\

\end{bmatrix}}$

decoupling or parallelization of MIMO channelAll TX symbols are now decoupled and paralelized

$\left. \mspace{20mu} \begin{matrix}{\overset{\sim}{y} = {{\sigma_{1}{\overset{\sim}{x}}_{1}} + {\overset{\sim}{n}}_{1}}} \\{\overset{\sim}{y} = {{\sigma_{2}{\overset{\sim}{x}}_{2}} + {\overset{\sim}{n}}_{2}}} \\{\vdots \vdots \vdots \vdots} \\{{\overset{\sim}{y}}_{t} = {{\sigma_{t}{\overset{\sim}{x}}_{t}} + {\overset{\sim}{n}}_{t}}}\end{matrix} \right\} \mspace{20mu} {Collection}\mspace{14mu} {of}\mspace{14mu} t\text{-}{parallel}\mspace{14mu} {{channels}.\mspace{14mu} {Transmitting}}\mspace{14mu} t\text{-}{information}\mspace{14mu} {symbols}\mspace{14mu} {in}\mspace{14mu} {parallel}\mspace{14mu} {Spatial}\mspace{14mu} {{multiplexity}.}${tilde over ({right arrow over (n)})}=U ^(H) {right arrow over (n)}

noise covariance=E{ññ ^(H) }=E{U ^(H) nn ^(H) U}=U ^(H)σ_(n) ²1U=σ _(n)² U ^(H) U=σ _(n) ²1

Noise is uncorrelated from different paths. That is the power of thenoise before and after beam forming is the same.

$\sigma_{\overset{\sim}{n}}^{2} = {{\sigma_{n}^{2}\mspace{14mu} {SNR}\mspace{14mu} {of}\mspace{14mu} i^{th}\mspace{14mu} {parallel}} = {\frac{\sigma_{i}^{2}P_{i}}{\sigma_{n}^{2}}\mspace{14mu} {channel}}}$P ₁σ_(t)(gain)ñ _(t)(noise)

{tilde over (x)} ₁ →⊗→⊕→{tilde over (y)} ₁

{tilde over (x)} ₂ →⊗→⊕→{tilde over (y)} ₂ t-parallel channels

{tilde over (x)} _(t) →⊗→⊕→{tilde over (y)} _(t)

spatial multiplexing

${{SNR}\mspace{14mu} {of}\mspace{14mu} {the}\mspace{14mu} i^{th}\mspace{14mu} {stream}} = \frac{P_{i}\sigma_{i}^{2}}{\sigma_{n}^{2}}$$\frac{{maximum}\mspace{14mu} {rate}}{Bandwidth} = {\frac{{Shannon}\mspace{14mu} {capacity}}{Bandwidth} = {\log_{2}\left( {1 + {SNR}} \right)}}$${{capacity}\mspace{14mu} {of}\mspace{14mu} i^{th}\mspace{14mu} {parallel}\mspace{14mu} {channel}} = {\log_{2}\left( {1 + \frac{P_{i}\sigma_{i}^{2}}{\sigma_{n}^{2}}} \right)}$

capacity of paths

$c_{1} = {\log_{2}\left( {1 + \frac{P_{1}\sigma_{1}^{2}}{\sigma_{n}^{2}}} \right)}$$c_{2} = {\log_{2}\left( {1 + \frac{P_{2}\sigma_{2}^{2}}{\sigma_{n}^{2}}} \right)}$$c_{t} = {\log_{2}\left( {1 + \frac{P_{t}\sigma_{t}^{2}}{\sigma_{n}^{2}}} \right)}$

Total MIMO Capacity=Σc_(i)

$c_{total} = {\sum\limits_{i = 1}^{t}{\log_{2}\left( {1 + \frac{P_{i}\sigma_{i}^{2}}{\sigma_{n}^{2}}} \right)}}$

sum of individual capacities of each parallel stream (t-informationstream)

Allocation of Power Optimally to Maximize Capacity

Given a transmit power P₀, how to optimally allocate P_(i) to alltransmitters.

P ₀≥Σ_(i) P _(i)

maximize capacity

${{Max}\mspace{14mu} c} = {{Max}\left\lbrack {\sum\limits_{i}^{t}{\log_{2}\left( {1 + \frac{P_{i}\sigma_{i}^{2}}{\sigma_{n}^{2}}} \right)}} \right\rbrack}$${{subject}\mspace{14mu} {to}\mspace{14mu} {\sum\limits_{i = 1}^{t}P_{i}}} = P_{0}$

Constraint Maximization problem (Lagrange multiplier)

$F = {{\sum\limits_{i = 1}^{t}{\log_{2}\left( {1 + \frac{P_{i}\sigma_{i}^{2}}{\sigma_{n}^{2}}} \right)}} + {\lambda \left( {P_{0} - {\Sigma \; P_{i}}} \right)}}$$\frac{dF}{{dP}_{i}} = 0$$P_{i} = {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{i}^{2}}}$${\frac{\sigma_{1}^{2}/\sigma_{n}^{2}}{1 + \frac{P_{1}\sigma_{1}^{2}}{\sigma_{n}^{2}}} + {\lambda \left( {- 1} \right)}} = 0$$\frac{\sigma_{1}^{2}/\sigma_{n}^{2}}{1 + \frac{\sigma_{1}^{2}}{\sigma_{n}^{2}}} = \lambda$${P_{1}\frac{\sigma_{1}^{2}}{\sigma_{n}^{2}}\frac{1}{\lambda}} = {1 + \frac{\sigma_{1}^{2}}{\sigma}}$$P_{1} = {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{1}^{2}}}$$\frac{1}{\lambda} = {\frac{\sigma_{n}^{2}}{\sigma_{1}^{2}} + P_{1}}$$P_{1} = \left( {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{1}^{2}}} \right)^{+}$$P_{2} = \left( {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{2}^{2}}} \right)^{+}$⋮$P_{t} = \left( {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{t}^{2}}} \right)^{+}$

power can only be positive (+)

x ⁺ x if x≥0 0 if x<0

${\sum\limits_{i = 1}^{t}P_{i}} = P_{0}$${\sum\limits_{i = 1}^{t}\left( {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{i}^{2}}} \right)^{+}} = P_{0}$water  filling$P_{i} = {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{i}^{2}}}$σ₁ = largest

Computational Method

Assume all N=t channels have non-zero or positive power

${{\frac{1}{\lambda} \geq {\frac{\sigma_{n}^{2}}{\sigma_{i}^{2}}\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} i}} = 1},2,{\ldots \mspace{14mu} N}$${\sum\limits_{i = 1}^{t}\left( {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{i}^{2}}} \right)} = P_{0}$σ₁ > σ₂ > σ₃  …   > 0

check

$P_{N} = {\frac{1}{\lambda} - \frac{\sigma_{n}^{2}}{\sigma_{N}^{2}}}$

if P_(N)>0 then procedure terminates

Asymptotic Capacity

$\begin{matrix}{{c = {\log_{2}{{1 + {\frac{1}{\sigma_{n}^{2}}{HR}_{xx}H^{H}}}}}}{R_{xx} = {E\left\lbrack {xx}^{H} \right\rbrack}}{R_{xx} = {\frac{P_{0}}{t}1}}{c = {\log_{2}{{1 + {\frac{P_{0}}{t\; \sigma_{n}^{2}}{HH}^{H}}}}}}{{{assume}\mspace{14mu} t}{> >}{r\mspace{14mu} {more}\mspace{14mu} {columns}\mspace{14mu} {than}\mspace{14mu} {rows}}}{{HH}^{H} = {{\begin{bmatrix}h_{1}^{H} \\h_{2}^{H} \\\vdots \\h_{t}^{H}\end{bmatrix}\begin{bmatrix}h_{1} & h_{2} & \ldots & h_{r}\end{bmatrix}} = \begin{bmatrix}{h_{1}^{H}h_{1}} & {h_{1}^{H}h_{2}} & \ldots \\{h_{2}^{H}h_{1}} & \ddots & \; \\\vdots & \; & {h_{r}^{H}h_{r}}\end{bmatrix}}}{{h_{i}^{H}h_{i}} = \left. {h_{i}}^{2}\rightarrow t \right.}\left. {h_{i}^{H}h_{j}}\rightarrow{{0\mspace{14mu} {when}\mspace{14mu} i} \neq j} \right.{\left. {HH}^{H}\rightarrow\begin{bmatrix}t & 0 & \ldots & 0 \\0 & t & \; & \; \\\vdots & \; & \ddots & \; \\\; & \; & \; & t\end{bmatrix} \right. = {t\; 1_{r}}}} & \; \\\begin{matrix}{c = {\log_{2}{{1 + {\frac{P_{0}}{t\; \sigma_{n}^{2}}t\; 1}}}}} \\{= {\log_{2}{{1 + {\frac{P_{0}}{\sigma_{n}^{2}}1}}}}} \\{= {\log_{2}{{1 + {\frac{P_{0}}{\sigma_{n}^{2}}1}}}}} \\{= {\log_{2}{\begin{matrix}{1 + \frac{P_{0}}{\sigma_{n}^{2}}} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & {1 + \frac{P_{0}}{\sigma_{n}^{2}}}\end{matrix}}}} \\{{= {\log_{2}\left( {1 + \frac{P_{0}}{\sigma_{n}^{2}}} \right)}^{r}}{{c_{asymptotic} = {r\; {\log_{2}\left( {1 + \frac{P_{0}}{\sigma_{n}^{2}}} \right)}}}\mspace{20mu} {constant}\mspace{14mu} {TX}\mspace{14mu} {Power}}{c_{asymptotic} \propto {{f(r)}\mspace{14mu} {linear}\mspace{14mu} {function}\mspace{14mu} {of}\mspace{14mu} {receivers}}}{c_{asymptotic} = {{\min \left( {r,t} \right)}{\log_{2}\left( {1 + \frac{P_{0}}{\sigma_{n}^{2}}} \right)}}}}\end{matrix} & \;\end{matrix}$

Space-time Alamouti code:

  1_(RX) × 2_(tx)  system  2tx  1RX$\mspace{20mu} {H = \begin{bmatrix}h_{1} & h_{2}\end{bmatrix}}$ $\mspace{20mu} {y = {{\begin{bmatrix}h_{1} & h_{2}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} + \overset{\rightarrow}{n}}}$$\mspace{20mu} {{{pre}\text{-}{code}\mspace{14mu} x_{1}} = {\frac{h_{1}^{*}}{h}x}}$$\mspace{20mu} {x_{2} = {{\frac{h_{2}^{*}}{h}{x\mspace{20mu}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}}} = {\begin{bmatrix}\frac{h_{1}^{*}}{h} \\\frac{h_{2}^{*}}{h}\end{bmatrix}x}}}$ $\mspace{20mu} {y = {{{\begin{bmatrix}h_{1} & h_{2}\end{bmatrix}\begin{bmatrix}\frac{h^{*}}{h} \\\frac{h_{2}^{*}}{h}\end{bmatrix}}x} + \overset{\rightarrow}{n}}}$$\mspace{20mu} {y = {{{h}x} + \overset{\rightarrow}{n}}}$${SNR} = \left. \frac{{h}^{2}P_{0}}{\sigma_{n}^{2}}\rightarrow{{exactly}\mspace{14mu} {the}\mspace{14mu} {same}\mspace{14mu} {as}\mspace{14mu} {MRC}\mspace{14mu} {similar}\mspace{14mu} {to}\mspace{14mu} {RX}\mspace{14mu} {diversity}} \right.$

Transmit vector

$= {\begin{bmatrix}\frac{h_{1}^{*}}{h} \\\frac{h_{2}^{*}}{h}\end{bmatrix}x\mspace{14mu} {can}\mspace{14mu} {this}\mspace{14mu} {be}\mspace{14mu} {done}\mspace{14mu} {at}\mspace{14mu} {{TX}?}}$

We need knowledge of h1 and h2 at TX

Channel State Information (CSI)

Therefore TX-beam forming is only possible when channel stateinformation is available at TX!Not always possible. Hence, obtaining TX diversity is challengingcompared to obtaining RX diversity.

Alamouti-code

1. Space-time code for 1_(RX)×2_(tx)2. Achieves a diversity of order 2 without CSI at TX

Orthogonal Space-Time Block Codes (OSTBC) Alamouti Code:

1_(RX)×2_(TX) we do not know h1, h2

Consider 2 symbols

$\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}\quad$

-   -   From TX1        First time instant symbols transmit

$\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}\quad$

-   -   From TX2

${y(1)} = {{\begin{bmatrix}h_{1} & h_{2}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} + {n(1)}}$

-   -   first TX incident        second time transmit instant        transmit

$\begin{bmatrix}{- x_{2}^{*}} \\x_{1}^{*}\end{bmatrix}$ ${y(2)} = {{\begin{bmatrix}h_{1} & h_{2}\end{bmatrix}\begin{bmatrix}{- x_{2}^{*}} \\x_{1}^{*}\end{bmatrix}} + {n(2)}}$ ${y^{*}(2)} = {{\begin{bmatrix}h_{1}^{*} & h_{2}^{*}\end{bmatrix}\begin{bmatrix}{- x_{2}} \\x_{1}\end{bmatrix}} + {n^{*}(2)}}$${y^{*}(2)} = {{\left\lbrack {- \begin{matrix}h_{1}^{*} & h_{2}^{*}\end{matrix}} \right\rbrack \begin{bmatrix}x_{2} \\x_{1}\end{bmatrix}} + {n^{*}(2)}}$ ${y^{*}(2)} = {{\begin{bmatrix}h_{2}^{*} & {- h_{1}^{*}}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} + {n^{*}(2)}}$

Now stack y(1) and y*(2)

$\begin{bmatrix}{y_{1}(1)} \\{y_{2}^{*}(2)}\end{bmatrix} = {{\begin{bmatrix}h_{1} & h_{2} \\h_{2}^{*} & {- h_{1}^{*}}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} + \begin{bmatrix}n_{1} \\n_{2}\end{bmatrix}}$

converted this to a MIMO system 2×2

1×2⇒2×2

c ₁ ^(H) c ₂ =h ₁ *h ₂ −h ₂ h ₁*=0c 1,c 2 column 1 and column 2 areorthogonal

Now what we do at receiver

$\frac{c_{1}}{c_{1}} = {w_{1} = {\begin{bmatrix}\frac{h_{1}}{h} \\\frac{h_{2}^{*}}{h}\end{bmatrix} = {\frac{1}{h}\begin{bmatrix}h_{1} \\h_{2}^{*}\end{bmatrix}}}}$

employ this as RX-beam forming

$\frac{c_{1}}{c_{1}}$

$\begin{matrix}{{w_{1}^{H}\overset{\rightarrow}{y}} = {{{\begin{bmatrix}\frac{h_{1}^{*}}{h} & \frac{h_{2}}{h}\end{bmatrix}\begin{bmatrix}h_{1} & h_{2} \\h_{2}^{*} & {- h_{1}^{*}}\end{bmatrix}}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} + {w_{1}^{H}n}}} \\{= {{\begin{bmatrix}{h} & 0\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} + \overset{\sim}{n}}} \\{= {{{h}x_{1}} + {\overset{\sim}{n}}_{1}}}\end{matrix}$

${SNR} = \frac{{h}^{2}P_{1}}{\sigma_{n}^{2}}$

diversity order 2 ∥h∥²=√{square root over (|h₁|²+|h₂|²)}similarly to decode x2, the beamformer w2 is

$w_{2} = {\frac{c_{2}}{c_{2}} = {\frac{1}{h}\begin{bmatrix}h_{2} \\{- h_{1}^{*}}\end{bmatrix}}}$

total TX P is fixed P0

${{TX}\mspace{14mu} {vector}} = \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}$ $P_{1} = {P_{2} = \frac{P_{0}}{2}}$

power is divided between 2 TXs

${SNR} = {{\frac{P_{0}}{2}\frac{{h}^{2}}{\sigma_{n}^{2}}} = {\frac{1}{2}\frac{{h}^{2}P}{\sigma_{n}^{2}}}}$

-   -   3 dB loss in SNR        Alamouti diversity gain sacrifices 3 dB to SNR, but it does not        require knowledge of the channel.        Alamouti belongs to orthogonal space-time block code (OSTBC)

$\begin{matrix}\; & \left. {space}\downarrow \right.\end{matrix}\underset{{time}\;\rightarrow}{\begin{matrix}x_{1} & {- x_{2}^{*}} \\x_{2} & x_{1}^{*}\end{matrix}}$

hence it transmits 2 symbols x₁, x₂ in 2 time instants. It effectivelytransmits 1 symbol/time instant. Therefore, it is a Rate R=1 code orFull Rate Code.

Another OSTBC:

1_(RX)×3_(TX) MIMO or MISO

channel matrix

[h₁ h₂ h₃] x₁ x₂ x₃ x₄

1^(st) to 8^(th) time instants

$\left. {\begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}\begin{matrix}{- x_{2}} & {- x_{3}} & {- x_{4}} & x_{1}^{*} & {- x_{2}^{*}} & {- x_{3}^{*}} & {- x_{4}^{*}} \\x_{1} & x_{4} & {- x_{3}} & x_{2}^{*} & x_{1}^{*} & x_{4}^{*} & {- x_{3}^{*}} \\x_{4} & x_{1} & x_{2} & x_{3}^{*} & {- x_{4}^{*}} & x_{1}^{*} & x_{2}\end{matrix}} \right\rbrack$

4 symbols over 8 time instants

${rate} = {\frac{4}{8} = {{\frac{1}{2}\mspace{31mu} R} = {\frac{1}{2}{code}}}}$${y(1)} = {{\begin{bmatrix}h_{1} & h_{2} & h_{3}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}} = {\begin{bmatrix}h_{1} & h_{2} & h_{3} & h_{4}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\0\end{bmatrix}}}$ h₁x₁ + h₂x₂ + h₃x₄ + 0 x₄ ${\begin{matrix}{{y(2)} = {\begin{bmatrix}h_{1} & h_{2} & h_{3}\end{bmatrix}\begin{bmatrix}{- x_{2}} \\x_{1} \\{- x_{4}}\end{bmatrix}}} \\{= {{{- h_{1}}x_{2}} + {h_{2}x_{1}} - {h_{3}x_{4}} + {0\; x_{3}}}} \\{= {{h_{2}x_{1}} + {\left( {- h_{1}} \right)x_{2}} + {0\; x_{3}} - {h_{3}x_{4}}}} \\{= {\begin{bmatrix}h_{2} & {- h_{1}} & 0 & {- h_{3}}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4}\end{bmatrix}}}\end{matrix}\begin{bmatrix}{y(1)} \\{y(2)} \\{y(3)} \\{y(4)} \\{y^{*}(5)} \\{y^{*}(6)} \\{y^{*}(7)} \\{y^{*}(8)}\end{bmatrix}} = {\begin{bmatrix}h_{1} & h_{2} & h_{3} & 0 \\h_{2} & {- h_{1}} & 0 & {- h_{3}} \\h_{3} & 0 & {- h_{1}} & h_{2} \\0 & h_{3} & {- h_{2}} & {- h_{1}} \\h_{1}^{*} & h_{2}^{*} & h_{3}^{*} & 0 \\h_{2}^{*} & {- h_{1}^{*}} & 0 & {- h_{3}^{*}} \\h_{3}^{*} & 0 & {- h_{1}^{*}} & h_{2}^{*} \\0 & h_{3}^{*} & {- h_{2}^{*}} & h_{1}^{*}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4}\end{bmatrix}}$ ${c_{1}^{H}c_{2}} = {{\begin{bmatrix}h_{1}^{*} & h_{2}^{*} & h_{3}^{*} & 0 & h_{1} & h_{2} & h_{3} & 0\end{bmatrix}\begin{bmatrix}h_{2} \\{- h_{1}} \\0 \\h_{3} \\h_{2}^{*} \\h_{1} \\0 \\h_{3}^{*}\end{bmatrix}} = 0}$ columns  are   orthogonal(OSTBC${Rate} = \frac{1}{2}$

Non-Linear MIMO Receiver V-BLAST Vertical Bell Labs Layered Space TimeArchitecture Successive Interference Cancellation (SIC)

Impact of each estimated symbol is cancelled. Further, because itemploys SIC, it is non-linear

$\begin{matrix}{\overset{\rightarrow}{y} = {{{H\overset{\rightarrow}{x}} + {n\mspace{31mu} r}} \geq t}} \\{= {{\begin{bmatrix}h_{1} & \ldots & h_{t}\end{bmatrix}\begin{bmatrix}x_{1} \\\vdots \\x_{t}\end{bmatrix}} + \overset{\rightarrow}{n}}}\end{matrix}$$\overset{\rightarrow}{y} = {{h_{1}x_{1}} + {h_{2}x_{2}} + {h_{3}x_{3}} + {\ldots \mspace{14mu} h_{t}x_{t}} + n_{1} + n_{2} + {\ldots \mspace{14mu} n_{t}}}$

consider pseudo-inverse or left-inverse of H

$\mspace{20mu} {Q = {H^{\dagger} = {{\begin{bmatrix}q_{1}^{H} \\q_{2}^{H} \\\vdots \\q_{t}^{H}\end{bmatrix}\mspace{20mu} {QH}} = {{1\mspace{20mu} H^{\dagger}H} = 1}}}}$${QH} = {1 = {{\begin{bmatrix}q_{1}^{H} \\q_{2}^{H} \\\vdots \\q_{t}^{H}\end{bmatrix}\begin{bmatrix}h_{1} & h_{2} & \ldots & h_{t}\end{bmatrix}} = {1_{t} = {\begin{bmatrix}{q_{1}^{H}h_{1}} & {q_{1}^{H}h_{2}} & {\ldots \mspace{14mu} q} \\{q_{2}^{H}h_{1}} & {q_{2}^{H}h_{2}} & \ldots \\\vdots & \; & \ddots\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & \ddots\end{bmatrix}}}}}$q ₁ ^(H) h ₁ =q ₂ ^(H) h ₂= . . . 1 diagonal element

q ₁ ^(H) h ₂ =q ₂ ^(H) h ₁= . . . =0 off diagonal element

${q_{i}^{H}h_{j}} = \left\{ \begin{matrix}1 & {i = j} \\0 & {i \neq j}\end{matrix} \right.$y=h ₁ x ₁ +h ₂ x ₂ + . . . +h _(t) x _(t) +{right arrow over (n)}

left multiply

=q ₁ ^(H) {right arrow over (y)}

=q ₁ ^(H)(h ₁ x ₁ +h ₂ x ₂ + . . . +h _(t) x _(t))+q ₁ ^(H) n

=x ₁+0+ . . . +ñ

zero-forcing RX

=x₁+ñ now employed to decode x₁Now remove the effect of x₁ from receiver

Subtract

${\overset{\rightarrow}{y} - {h_{1}x_{1}}} = {\left( {{h_{1}x_{1}} + {h_{2}x_{2}} + {\ldots \mspace{14mu} h_{t}x_{t}}} \right) + \overset{\sim}{n} - {h_{1}x_{1}}}$$= {{{h_{2}x_{2}} + \ldots + {h_{t}x_{t}}} = {{\begin{bmatrix}h_{2} & h_{3} & \ldots & h_{t}\end{bmatrix}\begin{bmatrix}x_{2} \\x_{3} \\\vdots \\x_{t}\end{bmatrix}} + \overset{\rightarrow}{n}}}$ r × (t − 1)  matrix

By cancelling x₁, this is effectively reduced to a rx(t−1) MIMO system.

$y_{2} = {{H^{1}\begin{bmatrix}x_{2} \\\vdots \\x_{t}\end{bmatrix}} + \overset{\rightarrow}{n}}$

consider Q¹=(H¹)^(†)Now repeat the process by decoding x2 and so onSuccessively cancelling . . .The advantage of this process or system is diversity order progressivelyincreases as the process proceeds through the scheme.

y _(t) =h _(t) x _(t) +n _(t)

Streams that are decoded later experience progressively higheraccuracy(higher diversity) of detection and lower BER. A non-linearreceiver.

MIMO Beamforming

Beamforming in the context of MIMO, implies transmission in one spatialdimension.

$\begin{matrix}{y = {{{Hx} + {\overset{\_}{n}\mspace{14mu} r}} \geq t}} \\{= {{U\; \Sigma \; V^{H}x} + \overset{\_}{n}}} \\{= {{{{\begin{bmatrix}u_{1} & u_{2} & \ldots & u_{t}\end{bmatrix}\begin{bmatrix}\sigma_{1} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \sigma_{t}\end{bmatrix}}\begin{bmatrix}v_{1}^{H} \\\vdots \\v_{t}^{H}\end{bmatrix}}x} + \overset{\_}{n}}}\end{matrix}$

transmit vector x=v₁

transmitting one symbol {tilde over (x)}₁dominant transmission mode of MIMO in an abstract space in an abstractdirection.

$\overset{\_}{y} = {{{{\begin{bmatrix}{\overset{\_}{u}}_{1} & {\overset{\_}{u}}_{2} & \ldots & {\overset{\_}{u}}_{t}\end{bmatrix}\begin{bmatrix}\sigma_{1} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \sigma_{t}\end{bmatrix}}\begin{bmatrix}v_{1}^{H} \\\vdots \\v_{t}^{H}\end{bmatrix}}v_{1}{\overset{\sim}{x}}_{1}} + \overset{\_}{n}}$$\overset{\_}{y} = {{{{\begin{bmatrix}{\overset{\_}{u}}_{1} & {\overset{\_}{u}}_{2} & \ldots & {\overset{\_}{u}}_{t}\end{bmatrix}\begin{bmatrix}\sigma_{1} & \; & \; \\\; & \ddots & \; \\\; & \; & \sigma_{t}\end{bmatrix}}\begin{bmatrix}1 \\0 \\\vdots \\0\end{bmatrix}}{\overset{\sim}{x}}_{1}} + \overset{\sim}{n}}$orthonormal  columns $\overset{\_}{y} = {{{\begin{bmatrix}{\overset{\_}{u}}_{1} & {\overset{\_}{u}}_{2} & \ldots & {\overset{\_}{u}}_{t}\end{bmatrix}\begin{bmatrix}\sigma_{1} \\\vdots \\0\end{bmatrix}}{\overset{\sim}{x}}_{1}} + \overset{\sim}{n}}$$\begin{matrix}{\overset{\_}{y} = {{\sigma_{1}{\overset{\_}{u}}_{1}{\overset{\sim}{x}}_{1}} + \overset{\sim}{n}}} \\{= {{\sigma_{1}{\overset{\sim}{x}}_{1}{\overset{\_}{u}}_{1}} + \overset{\_}{n}}}\end{matrix}$

at receiver, MRC can be determinedu₁ can be employed for MRC

u ₁ ^(H) y=u ₁ ^(H)(σ₁ ū ₁ {tilde over (x)} ₁ +{right arrow over(n)})=σ₁ {tilde over (x)} ₁ +u ₁ ^(H) n=σ ₁ {tilde over (x)} ₁ +ñ

${SNR} = \frac{\sigma_{1}^{2}P}{\sigma_{n}^{2}}$

σ₁=largest singular value=gain associated with dominant modeMaximal Ratio Transmission(MRT) results in a simplistic transmission andreception scheme for MIMO compared to MIMO-ZF, MIMO-MMSE, MIMO-VBLAST.MRT does beam forming but no higher throughput and no parallel channels.

MRT

1. MRT is capacity optimal for low SNRMRT achieves diversity order of rxt=high diversity

What is claimed is:
 1. A communications system, comprising: first signalprocessing circuitry for receiving a plurality of input data streams andapplying a different orthogonal function to each of the plurality ofinput data streams; second signal processing circuitry for processingeach of the plurality of input data streams having the differentorthogonal function applied thereto to multiplex a first group of theplurality of input data streams having a first group of orthogonalfunctions applied thereto onto a carrier signal and to multiplex asecond group of the plurality of input data streams having a secondgroup of orthogonal functions applied thereto onto the carrier signal; aMIMO transmitter for transmitting the carrier signal including the firstgroup of the plurality of input data streams having the first group oforthogonal functions applied thereto and the second group of theplurality of input data streams having the second group of orthogonalfunctions applied thereto over a plurality of separate communicationslinks, each of the plurality of separate communications links from onetransmitting antenna of a plurality of transmitting antennas to each ofa plurality of receiving antennas at a MIMO receiver.
 2. The system ofclaim 1, wherein the first signal processing circuitry furthercomprises: a carrier signal generator for generating a carrier signalfor each of the plurality of input data streams; a modulator formodulating each of the plurality of input data streams onto a separatecarrier signal; orthogonal processing circuitry for applying thedifferent orthogonal function to each of the separate carrier signals toprovide the plurality of input data streams having the differentorthogonal function applied thereto.
 3. The system of claim 1, furtherincluding: a receiver for receiving the carrier signal over theplurality of communications links through a plurality of receivingantennas; third signal processing circuitry for separating the firstgroup of the plurality of input data streams having the first group oforthogonal functions applied thereto from the second group of theplurality of input data streams having the second group of orthogonalfunctions applied thereto; and fourth signal processing circuitry forremoving the first and the second groups of orthogonal functions fromthe first and the second groups of the plurality of input data streamsrespectively.
 4. The system of claim 1, wherein the orthogonal functioncomprises at least one of orbital angular momentum function,Hermite-Gaussian function, Laguerre-Gaussian function, spatial Besselfunction, Prolate Spheroidal function.
 5. The system of claim 1, whereinthe MIMO transmitter transmits the carrier signal over at least one ofan optical link or a radio frequency (RF) link.
 6. The system of claim 1further comprising third signal processing circuitry for applyingmaximum ratio combining to the first group and the second group of theplurality of input data streams to improve a signal to noise ratio ofthe transmitted carrier signal.
 7. The system of claim 1, whereintransmitter transmits the carrier signal including the first and thesecond groups of the plurality of input data streams using a channelmatrix of an impulse response of a channel, the channel matrix iscreated using a pilot signal transmitted on a pilot channel.
 8. Thesystem of claim 7, wherein the channel matrix comprises a set ofsimultaneous equations, each equation representing a received signalwhich is a composite of a unique set of channel coefficients applied tothe transmitted carrier signal.
 9. The system of claim 7, wherein thechannel matrix can be decomposed using singular value decomposition. 10.A method for transmitting data over a communications link, comprisingreceiving a plurality of input data streams; applying a differentorthogonal function to each of the plurality of input data streams;multiplexing a first group of the plurality of input data streams havinga first group of orthogonal functions applied thereto onto a firstcarrier signal; multiplexing a second group of the plurality of inputdata streams having a second group of orthogonal functions appliedthereto onto a second carrier signal; temporally multiplexing the firstcarrier signal and the second carrier signal onto a third carriersignal; and transmitting the third carrier signal including the firstand the second groups of the plurality of input streams over a pluralityof communications links from a MIMO transmitter, each of thecommunications links transmitted from a separate transmitting antenna toa separate receiving antenna of a MIMO receiver.
 11. The method of claim10, wherein the step of applying further comprises: generating a carriersignal for each of the plurality of input data stream; modulating eachof the plurality of input data streams onto a separate carrier signal;applying the different orthogonal function to each of the separatecarrier signals to provide the plurality of input data streams havingthe different orthogonal function applied thereto.
 12. The method ofclaim 11, further including: receiving the carrier signal over theplurality of communications links through a plurality of receivingantennas; separating the first group of the plurality of input datastreams having the first orthogonal function applied thereto from thesecond group of the plurality of input data streams having the secondorthogonal function applied thereto; and removing the first and thesecond orthogonal function from the first and the second groups of theplurality of input data streams respectively.
 13. The method of claim10, wherein the orthogonal function comprises at least one of orbitalangular momentum function, Hermite-Gaussian function, Laguerre-Gaussianfunction, spatial Bessel function, Prolate Spheroidal function.
 14. Themethod of claim 10, wherein the step of transmitting further comprisestransmitting the carrier signal over at least one of an optical link ora radio frequency (RF) link.
 15. The method of claim 10 furthercomprising applying maximum ratio combining to the first group and thesecond group of the plurality of composite data streams to improve asignal to noise ratio of the transmitted carrier signal.
 16. The methodof claim 10, wherein the step of transmitting further comprises:creating a channel matrix using a pilot signal transmitted on a pilotchannel; and transmitting the carrier signal including the first and thesecond groups of the plurality of input data streams using the channelmatrix of an impulse response of a channel.
 17. The method of claim 16,wherein the channel matrix comprises a set of simultaneous equations,each equation representing a received signal which is a composite of aunique set of channel coefficients applied to the transmitted carriersignal.
 18. The method of claim 16 further comprising the step ofdecomposing the channel matrix using singular value decomposition.
 19. Acommunications system, comprising: a carrier signal generator forgenerating a carrier signal for each of a plurality of input datastreams; a modulator for modulating each of the plurality of input datastreams onto a separate carrier signal; orthogonal function processingcircuitry for applying a different orthogonal function to each of theplurality of input data streams to provide a plurality of composite datastreams having the different orthogonal function applied thereto; firstsignal processing circuitry for processing each of the plurality ofinput data streams having the different orthogonal function appliedthereto to multiplex a first group of the plurality of input datastreams having a first group of orthogonal functions applied theretoonto a carrier signal and to multiplex a second group of the pluralityof input data streams having a second group of orthogonal functionsapplied thereto onto the carrier signal; a MIMO transmitter fortransmitting the carrier signal including the first group of theplurality of input data streams having the first group of orthogonalfunctions applied thereto and the second group of the plurality of inputdata streams having the second group of orthogonal functions appliedthereto over a plurality of communications links each of the pluralityof separate communications links from one transmitting antenna of aplurality of transmitting antennas to each of a plurality of receivingantennas; a MIMO receiver including the plurality of receiving antennasfor receiving the carrier signal over each of the plurality ofcommunications links through the plurality of receiving antennas; secondsignal processing circuitry for demultiplexing the first group of theplurality of input data streams having the first group of orthogonalfunctions applied thereto from the second group of the plurality ofinput data streams having the second group of orthogonal functionsapplied thereto; and third signal processing circuitry for removing thefirst and the second group of orthogonal functions from the first andthe second groups of the plurality of input data streams respectively.20. The system of claim 19, wherein the orthogonal function comprises atleast one of orbital angular momentum function, Hermite-Gaussianfunction, Laguerre-Gaussian function, spatial Bessel function, ProlateSpheroidal function.
 21. The system of claim 19, wherein the MIMOtransmitter transmits the carrier signal over at least one of an opticallink or a radio frequency (RF) link.
 22. The system of claim 19 furthercomprising fourth signal processing circuitry for applying maximum ratiocombining to the first group and the second group of the plurality ofinput data streams to improve a signal to noise ratio of the transmittedcarrier signal.
 23. The system of claim 19, wherein the MIMO transmittertransmits the carrier signal including the first and the second groupsof the plurality of input data streams using a channel matrix of animpulse response of a channel, the channel matrix is created using apilot signal transmitted on a pilot channel.
 24. The system of claim 23,wherein the channel matrix comprises a set of simultaneous equations,each equation representing a received signal which is a composite of aunique set of channel coefficients applied to the transmitted carriersignal.
 25. The system of claim 23, wherein the channel matrix can bedecomposed using singular value decomposition.
 26. A communicationssystem, comprising: first signal processing circuitry for receiving theplurality of input data streams and applying a different orthogonalfunction to each of the plurality of input data streams; second signalprocessing circuitry for processing each of the plurality of input datastreams having the different orthogonal function applied thereto tomultiplex a first group of the plurality of input data streams having afirst group of orthogonal functions applied thereto onto a carriersignal and to multiplex a second group of the plurality of input datastreams having a second group of orthogonal functions applied theretoonto the carrier signal; third signal processing circuitry for applyingmaximum ratio combining to the carrier signal to improve a signal tonoise ratio of a transmitted carrier signal; and a MIMO transmitter fortransmitting the carrier signal including the plurality of input datastreams having the maximum ration combining applied thereto over aplurality of communications links, each of the plurality of separatecommunications links from one transmitting antenna a plurality oftransmitting antennas to each of a plurality of receiving antennas at aMIMO receiver.
 27. The system of claim 26, further including: a receiverfor receiving the carrier signal over the plurality of communicationslinks through the plurality of receiving antennas; and second signalprocessing circuitry for processing the received carrier signal usingmaximum ratio combining and outputting the plurality of input datastreams.
 28. The system of claim 26, wherein transmitter transmits thecarrier signal over at least one of an optical link or a radio frequencylink.
 29. The system of claim 26, wherein transmitter transmits thecarrier signal including the first and second groups of the plurality ofinput data streams using a channel matrix of an impulse response of achannel, the channel matrix is created using a pilot signal transmittedon a pilot channel.
 30. The system of claim 29, wherein the channelmatrix comprises a set of simultaneous equations, each equationrepresenting a received signal which is a composite of a unique set ofchannel coefficients applied to the transmitted carrier signal.